Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.1% → 91.8%

Time: 9.9s

Alternatives: 22

Speedup: 6.6×

• 28.0% 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: 55.1% 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: 91.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_m}{\ell} \cdot \left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \left(\mathsf{fma}\left(\frac{k}{t\_m}, \frac{k}{t\_m}, 2\right) \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

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 (<= t_m 3.1e-117)
    (*
     2.0
     (* l (/ (/ (* (cos k) l) (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) k)))
    (/
     2.0
     (*
      (/ t_m l)
      (*
       (* t_m (* (/ t_m l) (sin k)))
       (* (fma (/ k t_m) (/ k t_m) 2.0) (tan k))))))))

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 (t_m <= 3.1e-117) {
		tmp = 2.0 * (l * (((cos(k) * l) / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) / k));
	} else {
		tmp = 2.0 / ((t_m / l) * ((t_m * ((t_m / l) * sin(k))) * (fma((k / t_m), (k / t_m), 2.0) * tan(k))));
	}
	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 (t_m <= 3.1e-117)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) / k)));
	else
		tmp = Float64(2.0 / Float64(Float64(t_m / l) * Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(fma(Float64(k / t_m), Float64(k / t_m), 2.0) * tan(k)))));
	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[t$95$m, 3.1e-117], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000011e-117

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto 2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)}}{\color{blue}{k}}\right) \]

   if 3.10000000000000011e-117 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   9. Applied rewrites 78.9%

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

Alternative 2: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\ell} \cdot \sin k\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}}{k}\right)\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(\frac{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}{\ell} \cdot \left(t\_m \cdot t\_m\right)\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot t\_2\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (* (/ t_m l) (sin k))))
   (*
    t_s
    (if (<= t_m 3.1e-117)
      (*
       2.0
       (*
        l
        (/ (/ (* (cos k) l) (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) k)))
      (if (<= t_m 8e+146)
        (/
         2.0
         (*
          (* (/ (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) l) (* t_m t_m))
          t_2))
        (/ 2.0 (* (* (* (* t_m t_2) (/ t_m l)) (tan 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 t_2 = (t_m / l) * sin(k);
	double tmp;
	if (t_m <= 3.1e-117) {
		tmp = 2.0 * (l * (((cos(k) * l) / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) / k));
	} else if (t_m <= 8e+146) {
		tmp = 2.0 / ((((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) / l) * (t_m * t_m)) * t_2);
	} else {
		tmp = 2.0 / ((((t_m * t_2) * (t_m / l)) * tan(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)
	t_2 = Float64(Float64(t_m / l) * sin(k))
	tmp = 0.0
	if (t_m <= 3.1e-117)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) / k)));
	elseif (t_m <= 8e+146)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) / l) * Float64(t_m * t_m)) * t_2));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_2) * Float64(t_m / l)) * tan(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_] := Block[{t$95$2 = N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.1e-117], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+146], N[(2.0 / N[(N[(N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$2), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.10000000000000011e-117

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto 2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)}}{\color{blue}{k}}\right) \]

   if 3.10000000000000011e-117 < t < 7.99999999999999947e146

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 61.1%

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

   if 7.99999999999999947e146 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 3: 90.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.1 \cdot 10^{-90}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}}{k}\right)\\ \mathbf{elif}\;t\_m \leq 8 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot t\_m\right) \cdot \sin k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= t_m 2.1e-90)
    (*
     2.0
     (* l (/ (/ (* (cos k) l) (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) k)))
    (if (<= t_m 8e+146)
      (/
       2.0
       (*
        (* (* (* t_m t_m) (sin k)) (/ t_m l))
        (/ (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) l)))
      (/
       2.0
       (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan 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 (t_m <= 2.1e-90) {
		tmp = 2.0 * (l * (((cos(k) * l) / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) / k));
	} else if (t_m <= 8e+146) {
		tmp = 2.0 / ((((t_m * t_m) * sin(k)) * (t_m / l)) * ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) / l));
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(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 (t_m <= 2.1e-90)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) / k)));
	elseif (t_m <= 8e+146)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * sin(k)) * Float64(t_m / l)) * Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) / l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(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[t$95$m, 2.1e-90], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 8e+146], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.0999999999999999e-90

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto 2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)}}{\color{blue}{k}}\right) \]

   if 2.0999999999999999e-90 < t < 7.99999999999999947e146

   1. Initial program 55.1%

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

   3. Applied rewrites 59.3%

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

   if 7.99999999999999947e146 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 4: 90.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}}{k}\right)\\ \mathbf{elif}\;t\_m \leq 1.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot t\_m\right) \cdot \sin k\right) \cdot \left(\frac{t\_m}{\ell} \cdot \tan k\right)\right) \cdot \mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= t_m 1.1e-50)
    (*
     2.0
     (* l (/ (/ (* (cos k) l) (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) k)))
    (if (<= t_m 1.4e+147)
      (*
       (/
        2.0
        (*
         (* (* (* t_m t_m) (sin k)) (* (/ t_m l) (tan k)))
         (fma (/ k (* t_m t_m)) k 2.0)))
       l)
      (/
       2.0
       (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan 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 (t_m <= 1.1e-50) {
		tmp = 2.0 * (l * (((cos(k) * l) / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) / k));
	} else if (t_m <= 1.4e+147) {
		tmp = (2.0 / ((((t_m * t_m) * sin(k)) * ((t_m / l) * tan(k))) * fma((k / (t_m * t_m)), k, 2.0))) * l;
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(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 (t_m <= 1.1e-50)
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) / k)));
	elseif (t_m <= 1.4e+147)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * sin(k)) * Float64(Float64(t_m / l) * tan(k))) * fma(Float64(k / Float64(t_m * t_m)), k, 2.0))) * l);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(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[t$95$m, 1.1e-50], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.4e+147], N[(N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.0999999999999999e-50

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

       \[\leadsto 2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\right)}}{\color{blue}{k}}\right) \]

   if 1.0999999999999999e-50 < t < 1.4e147

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 59.0%

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

   if 1.4e147 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 5: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}}{k}\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     2.0
     (*
      l
      (/ (/ (* (cos k) l) (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) k))))))

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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 * (l * (((cos(k) * l) / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) / k));
	}
	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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) / k)));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

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

Alternative 6: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\frac{\cos k \cdot \ell}{k}}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)}\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     2.0
     (*
      l
      (/ (/ (* (cos k) l) k) (* (fma (cos (+ k k)) -0.5 0.5) (* k 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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 * (l * (((cos(k) * l) / k) / (fma(cos((k + k)), -0.5, 0.5) * (k * 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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(Float64(Float64(cos(k) * l) / k) / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m)))));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

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

Alternative 7: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\frac{\ell}{\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\cos k}{k}\right)\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     2.0
     (*
      l
      (* (/ l (* (fma (cos (+ k k)) -0.5 0.5) (* k t_m))) (/ (cos k) k)))))))

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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 * (l * ((l / (fma(cos((k + k)), -0.5, 0.5) * (k * t_m))) * (cos(k) / k)));
	}
	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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(Float64(l / Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * Float64(k * t_m))) * Float64(cos(k) / k))));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(N[(l / N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 69.1%

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

Alternative 8: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right) \cdot \left(k \cdot t\_m\right)}\right)\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     2.0
     (*
      l
      (* l (/ (cos k) (* (* (fma (cos (+ k k)) -0.5 0.5) k) (* k 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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / ((fma(cos((k + k)), -0.5, 0.5) * k) * (k * 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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k) * Float64(k * t_m))))));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

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

Alternative 9: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(t\_m \cdot \left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot k\right)\right) \cdot k}\right)\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     2.0
     (*
      l
      (* l (/ (cos k) (* (* t_m (* (fma (cos (+ k k)) -0.5 0.5) k)) k))))))))

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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = 2.0 * (l * (l * (cos(k) / ((t_m * (fma(cos((k + k)), -0.5, 0.5) * k)) * k))));
	}
	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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(2.0 * Float64(l * Float64(l * Float64(cos(k) / Float64(Float64(t_m * Float64(fma(cos(Float64(k + k)), -0.5, 0.5) * k)) * k)))));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(l * N[(l * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$m * N[(N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 66.0%

      \[\leadsto 2 \cdot \left(\ell \cdot \left(\ell \cdot \frac{\cos k}{\left(\left(\mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right) \cdot t\right) \cdot k\right) \cdot \color{blue}{k}}\right)\right) \]

   10. Applied rewrites 66.0%

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

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \ell}{\left(\left(k \cdot k\right) \cdot t\_m\right) \cdot \mathsf{fma}\left(\cos \left(k + k\right), -0.5, 0.5\right)} \cdot \left(\ell + \ell\right)\\ \end{array} \end{array} \]

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 10500000000.0)
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan k)) 2.0))
    (*
     (/ (* (cos k) l) (* (* (* k k) t_m) (fma (cos (+ k k)) -0.5 0.5)))
     (+ l l)))))

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 <= 10500000000.0) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(k)) * 2.0);
	} else {
		tmp = ((cos(k) * l) / (((k * k) * t_m) * fma(cos((k + k)), -0.5, 0.5))) * (l + l);
	}
	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 <= 10500000000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(k)) * 2.0));
	else
		tmp = Float64(Float64(Float64(cos(k) * l) / Float64(Float64(Float64(k * k) * t_m) * fma(cos(Float64(k + k)), -0.5, 0.5))) * Float64(l + l));
	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[k, 10500000000.0], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(N[Cos[N[(k + k), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.05e10

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

   if 1.05e10 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   6. Applied rewrites 62.7%

      \[\leadsto 2 \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{\cos k}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot k\right)\right)}\right)}\right) \]

   8. Applied rewrites 62.7%

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

Alternative 11: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot \sin k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot 2}\\ \end{array} \end{array} \]

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 (<= l 1.2e+119)
    (/
     2.0
     (*
      (* (* (* t_m (* (/ t_m l) k)) (/ t_m l)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
    (/ 2.0 (* (* (* (* t_m (* (/ t_m l) (sin k))) (/ t_m l)) (tan 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 (l <= 1.2e+119) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(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 (l <= 1.2d+119) then
        tmp = 2.0d0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    else
        tmp = 2.0d0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(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 (l <= 1.2e+119) {
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * Math.sin(k))) * (t_m / l)) * Math.tan(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 l <= 1.2e+119:
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	else:
		tmp = 2.0 / ((((t_m * ((t_m / l) * math.sin(k))) * (t_m / l)) * math.tan(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 (l <= 1.2e+119)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * k)) * Float64(t_m / l)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * sin(k))) * Float64(t_m / l)) * tan(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 (l <= 1.2e+119)
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	else
		tmp = 2.0 / ((((t_m * ((t_m / l) * sin(k))) * (t_m / l)) * tan(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[l, 1.2e+119], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $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], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e119

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

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

   if 1.2e119 < l

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 68.3%

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

Alternative 12: 74.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \left(\frac{t\_m}{\ell} \cdot k\right)\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 (<= t_m 6e-134)
    (* 2.0 (* (/ l (pow k 4.0)) (/ l t_m)))
    (/
     2.0
     (*
      (* (* (* t_m (* (/ t_m l) k)) (/ t_m l)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

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 (t_m <= 6e-134) {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	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 (t_m <= 6d-134) then
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t_m))
    else
        tmp = 2.0d0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    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 (t_m <= 6e-134) {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	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 t_m <= 6e-134:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t_m))
	else:
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	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 (t_m <= 6e-134)
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(t_m / l) * k)) * Float64(t_m / l)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	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 (t_m <= 6e-134)
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t_m));
	else
		tmp = 2.0 / ((((t_m * ((t_m / l) * k)) * (t_m / l)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	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[t$95$m, 6e-134], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6e-134

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.2%

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

   9. Applied rewrites 56.8%

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

   if 6e-134 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.8%

       \[\leadsto \frac{2}{\left(\left(\left(t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{k}\right)\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot \frac{k \cdot t\_m}{\ell}\right) \cdot \frac{t\_m}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)}\\ \end{array} \end{array} \]

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 (<= t_m 6e-134)
    (* 2.0 (* (/ l (pow k 4.0)) (/ l t_m)))
    (/
     2.0
     (*
      (* (* (* t_m (/ (* k t_m) l)) (/ t_m l)) (tan k))
      (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))))))

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 (t_m <= 6e-134) {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = 2.0 / ((((t_m * ((k * t_m) / l)) * (t_m / l)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0));
	}
	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 (t_m <= 6d-134) then
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t_m))
    else
        tmp = 2.0d0 / ((((t_m * ((k * t_m) / l)) * (t_m / l)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))
    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 (t_m <= 6e-134) {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = 2.0 / ((((t_m * ((k * t_m) / l)) * (t_m / l)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0));
	}
	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 t_m <= 6e-134:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t_m))
	else:
		tmp = 2.0 / ((((t_m * ((k * t_m) / l)) * (t_m / l)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))
	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 (t_m <= 6e-134)
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * Float64(Float64(k * t_m) / l)) * Float64(t_m / l)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t_m) ^ 2.0)) + 1.0)));
	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 (t_m <= 6e-134)
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t_m));
	else
		tmp = 2.0 / ((((t_m * ((k * t_m) / l)) * (t_m / l)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0));
	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[t$95$m, 6e-134], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m * N[(N[(k * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l), $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]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 6e-134

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.2%

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

   9. Applied rewrites 56.8%

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

   if 6e-134 < t

   1. Initial program 55.1%

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

   3. Applied rewrites 66.8%

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.0%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 70.6%

       \[\leadsto \frac{2}{\left(\left(\left(t \cdot \color{blue}{\frac{k \cdot t}{\ell}}\right) \cdot \frac{t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t\_m \cdot t\_m\right) \cdot k\right) \cdot \frac{t\_m}{\ell}\right) \cdot \frac{\mathsf{fma}\left(\frac{k}{t\_m \cdot t\_m}, k, 2\right) \cdot \tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t\_m}\right)\\ \end{array} \end{array} \]

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)))
       5e+303)
    (/
     2.0
     (*
      (* (* (* t_m t_m) k) (/ t_m l))
      (/ (* (fma (/ k (* t_m t_m)) k 2.0) (tan k)) l)))
    (* 2.0 (* (/ l (pow k 4.0)) (/ l 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 5e+303) {
		tmp = 2.0 / ((((t_m * t_m) * k) * (t_m / l)) * ((fma((k / (t_m * t_m)), k, 2.0) * tan(k)) / l));
	} else {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / 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 (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))) <= 5e+303)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m * t_m) * k) * Float64(t_m / l)) * Float64(Float64(fma(Float64(k / Float64(t_m * t_m)), k, 2.0) * tan(k)) / l)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m)));
	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], 5e+303], N[(2.0 / N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(k / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $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)))) < 4.9999999999999997e303

   1. Initial program 55.1%

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

   3. Applied rewrites 59.3%

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

   4. Taylor expanded in k around 0

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

   6. Applied rewrites 54.6%

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

   if 4.9999999999999997e303 < (/.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 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.2%

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

   9. Applied rewrites 56.8%

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

Alternative 15: 70.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.35 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t\_m \cdot t\_m}\right)}{t\_m}\\ \end{array} \end{array} \]

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 (<= t_m 1.35e-48)
    (* 2.0 (* (/ l (pow k 4.0)) (/ l t_m)))
    (/ (* (/ l k) (* (/ l k) (/ 1.0 (* t_m t_m)))) 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 (t_m <= 1.35e-48) {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / 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 (t_m <= 1.35d-48) then
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t_m))
    else
        tmp = ((l / k) * ((l / k) * (1.0d0 / (t_m * t_m)))) / 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 (t_m <= 1.35e-48) {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t_m));
	} else {
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / 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 t_m <= 1.35e-48:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t_m))
	else:
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / 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 (t_m <= 1.35e-48)
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m)));
	else
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64(t_m * t_m)))) / 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 (t_m <= 1.35e-48)
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t_m));
	else
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / 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[t$95$m, 1.35e-48], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.35000000000000006e-48

   1. Initial program 55.1%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 52.2%

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

   9. Applied rewrites 56.8%

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

   if 1.35000000000000006e-48 < t

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 66.0%

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

Alternative 16: 70.3% accurate, 0.8× speedup

Math

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

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       INFINITY)
    (/ (* (/ l k) (* (/ l k) (/ 1.0 (* t_m t_m)))) t_m)
    (/ (* (/ l t_m) (/ (/ l (* k k)) t_m)) 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= ((double) INFINITY)) {
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / t_m;
	} else {
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / t_m;
	}
	return t_s * tmp;
}

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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= Double.POSITIVE_INFINITY) {
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / t_m;
	} else {
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= math.inf:
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / t_m
	else:
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 (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))) <= Inf)
		tmp = Float64(Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(1.0 / Float64(t_m * t_m)))) / t_m);
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * k)) / t_m)) / 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= Inf)
		tmp = ((l / k) * ((l / k) * (1.0 / (t_m * t_m)))) / t_m;
	else
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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[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], Infinity], N[(N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $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)))) < +inf.0

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 66.0%

      \[\leadsto \frac{\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{t \cdot t}\right)}{t} \]

   if +inf.0 < (/.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 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 64.3%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\frac{\ell}{k \cdot k}}{t}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+19}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t\_m} \cdot \frac{\frac{\ell}{k \cdot k}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 (<=
       (/
        2.0
        (*
         (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
         (+ (+ 1.0 (pow (/ k t_m) 2.0)) 1.0)))
       1e+19)
    (* (/ l (* (* (* k t_m) (* t_m t_m)) k)) l)
    (/ (* (/ l t_m) (/ (/ l (* k k)) t_m)) 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+19) {
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	} else {
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+19) then
        tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l
    else
        tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+19) {
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	} else {
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+19:
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l
	else:
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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 (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+19)
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * t_m) * Float64(t_m * t_m)) * k)) * l);
	else
		tmp = Float64(Float64(Float64(l / t_m) * Float64(Float64(l / Float64(k * k)) / t_m)) / 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 1e+19)
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	else
		tmp = ((l / t_m) * ((l / (k * k)) / t_m)) / 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[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+19], N[(N[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(N[(N[(l / t$95$m), $MachinePrecision] * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $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)))) < 1e19

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 56.2%

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

   8. Applied rewrites 56.2%

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

   10. Applied rewrites 63.2%

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

   if 1e19 < (/.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 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 64.3%

      \[\leadsto \frac{\frac{\ell}{t} \cdot \frac{\frac{\ell}{k \cdot k}}{t}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{\left(t\_m \cdot t\_m\right) \cdot \left(k \cdot t\_m\right)} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t\_m}}{t\_m}}{t\_m}\\ \end{array} \end{array} \]

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 3e-18)
    (* (/ l (* (* t_m t_m) (* k t_m))) (/ l k))
    (/ (/ (/ (* l l) (* (* k k) t_m)) t_m) 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 <= 3e-18) {
		tmp = (l / ((t_m * t_m) * (k * t_m))) * (l / k);
	} else {
		tmp = (((l * l) / ((k * k) * t_m)) / t_m) / 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 <= 3d-18) then
        tmp = (l / ((t_m * t_m) * (k * t_m))) * (l / k)
    else
        tmp = (((l * l) / ((k * k) * t_m)) / t_m) / 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 <= 3e-18) {
		tmp = (l / ((t_m * t_m) * (k * t_m))) * (l / k);
	} else {
		tmp = (((l * l) / ((k * k) * t_m)) / t_m) / 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 <= 3e-18:
		tmp = (l / ((t_m * t_m) * (k * t_m))) * (l / k)
	else:
		tmp = (((l * l) / ((k * k) * t_m)) / t_m) / 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 <= 3e-18)
		tmp = Float64(Float64(l / Float64(Float64(t_m * t_m) * Float64(k * t_m))) * Float64(l / k));
	else
		tmp = Float64(Float64(Float64(Float64(l * l) / Float64(Float64(k * k) * t_m)) / t_m) / 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 <= 3e-18)
		tmp = (l / ((t_m * t_m) * (k * t_m))) * (l / k);
	else
		tmp = (((l * l) / ((k * k) * t_m)) / t_m) / 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, 3e-18], N[(N[(l / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(k * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l * l), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.99999999999999983e-18

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 56.2%

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

   8. Applied rewrites 64.4%

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

   if 2.99999999999999983e-18 < k

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 54.5%

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

   8. Applied rewrites 60.1%

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

Alternative 19: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) + 1\right)} \leq 10^{+19}:\\ \;\;\;\;\frac{\ell}{\left(\left(k \cdot t\_m\right) \cdot \left(t\_m \cdot t\_m\right)\right) \cdot k} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t\_m \cdot \left(t\_m \cdot \left(\left(k \cdot k\right) \cdot t\_m\right)\right)}\\ \end{array} \end{array} \]

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+19)
    (* (/ l (* (* (* k t_m) (* t_m t_m)) k)) l)
    (* l (/ l (* t_m (* t_m (* (* k k) 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 ((2.0 / ((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) + 1.0))) <= 1e+19) {
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	} else {
		tmp = l * (l / (t_m * (t_m * ((k * k) * 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 ((2.0d0 / (((((t_m ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) + 1.0d0))) <= 1d+19) then
        tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l
    else
        tmp = l * (l / (t_m * (t_m * ((k * k) * 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 ((2.0 / ((((Math.pow(t_m, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+19) {
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	} else {
		tmp = l * (l / (t_m * (t_m * ((k * k) * 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 (2.0 / ((((math.pow(t_m, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t_m), 2.0)) + 1.0))) <= 1e+19:
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l
	else:
		tmp = l * (l / (t_m * (t_m * ((k * k) * 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 (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+19)
		tmp = Float64(Float64(l / Float64(Float64(Float64(k * t_m) * Float64(t_m * t_m)) * k)) * l);
	else
		tmp = Float64(l * Float64(l / Float64(t_m * Float64(t_m * Float64(Float64(k * k) * 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 ((2.0 / (((((t_m ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) + 1.0))) <= 1e+19)
		tmp = (l / (((k * t_m) * (t_m * t_m)) * k)) * l;
	else
		tmp = l * (l / (t_m * (t_m * ((k * k) * 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[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+19], N[(N[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision], N[(l * N[(l / N[(t$95$m * N[(t$95$m * N[(N[(k * k), $MachinePrecision] * t$95$m), $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)))) < 1e19

   1. Initial program 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 56.2%

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

   8. Applied rewrites 56.2%

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

   10. Applied rewrites 63.2%

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

   if 1e19 < (/.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 55.1%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 51.9%

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

   6. Applied rewrites 56.2%

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

   8. Applied rewrites 62.7%

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

Alternative 20: 63.2% accurate, 6.6× speedup

Math

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

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 (* (/ l (* (* (* k t_m) (* t_m t_m)) k)) l)))

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 * ((l / (((k * t_m) * (t_m * t_m)) * k)) * l);
}

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 * t_m) * (t_m * t_m)) * k)) * l)
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 * ((l / (((k * t_m) * (t_m * t_m)) * k)) * l);
}

Python

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

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 / Float64(Float64(Float64(k * t_m) * Float64(t_m * t_m)) * k)) * l))
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 * t_m) * (t_m * t_m)) * k)) * l);
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[(l / N[(N[(N[(k * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 51.9%

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

6. Applied rewrites 56.2%

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

8. Applied rewrites 56.2%

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

10. Applied rewrites 63.2%

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

Alternative 21: 59.0% accurate, 6.6× speedup

Math

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

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 (* (/ l (* (* (* k k) t_m) (* t_m t_m))) l)))

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 * ((l / (((k * k) * t_m) * (t_m * t_m))) * l);
}

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 * k) * t_m) * (t_m * t_m))) * l)
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 * ((l / (((k * k) * t_m) * (t_m * t_m))) * l);
}

Python

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

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 / Float64(Float64(Float64(k * k) * t_m) * Float64(t_m * t_m))) * l))
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 * k) * t_m) * (t_m * t_m))) * l);
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[(l / N[(N[(N[(k * k), $MachinePrecision] * t$95$m), $MachinePrecision] * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 51.9%

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

6. Applied rewrites 56.2%

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

8. Applied rewrites 56.2%

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

10. Applied rewrites 59.0%

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

Alternative 22: 56.2% accurate, 6.6× speedup

Math

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

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 (* (/ l (* (* k k) (* (* t_m t_m) t_m))) l)))

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 * ((l / ((k * k) * ((t_m * t_m) * t_m))) * l);
}

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 * k) * ((t_m * t_m) * t_m))) * l)
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 * ((l / ((k * k) * ((t_m * t_m) * t_m))) * l);
}

Python

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

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 / Float64(Float64(k * k) * Float64(Float64(t_m * t_m) * t_m))) * l))
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 * k) * ((t_m * t_m) * t_m))) * l);
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[(l / N[(N[(k * k), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

2. Taylor expanded in k around 0

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

4. Applied rewrites 51.9%

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

6. Applied rewrites 56.2%

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

8. Applied rewrites 56.2%

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

Reproduce

herbie shell --seed 2025161 
(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))))