Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.4% → 96.3%

Time: 20.4s

Alternatives: 25

Speedup: 12.2×

• 39.3% 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: 35.4% 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: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{k}{\cos k}\\ t_3 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{-209}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_3 \cdot t\_2\right) \cdot \left(\left(\sin k \cdot k\right) \cdot t\_m\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_2 \cdot t\_3\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ k (cos k))) (t_3 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 4.6e-209)
      (/ 2.0 (/ (* (* t_3 t_2) (* (* (sin k) k) t_m)) l))
      (/ 2.0 (* (* t_2 t_3) (* k (/ (* (sin k) 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) {
	double t_2 = k / cos(k);
	double t_3 = sin(k) / l;
	double tmp;
	if (t_m <= 4.6e-209) {
		tmp = 2.0 / (((t_3 * t_2) * ((sin(k) * k) * t_m)) / l);
	} else {
		tmp = 2.0 / ((t_2 * t_3) * (k * ((sin(k) * t_m) / l)));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = k / cos(k)
    t_3 = sin(k) / l
    if (t_m <= 4.6d-209) then
        tmp = 2.0d0 / (((t_3 * t_2) * ((sin(k) * k) * t_m)) / l)
    else
        tmp = 2.0d0 / ((t_2 * t_3) * (k * ((sin(k) * t_m) / l)))
    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 t_2 = k / Math.cos(k);
	double t_3 = Math.sin(k) / l;
	double tmp;
	if (t_m <= 4.6e-209) {
		tmp = 2.0 / (((t_3 * t_2) * ((Math.sin(k) * k) * t_m)) / l);
	} else {
		tmp = 2.0 / ((t_2 * t_3) * (k * ((Math.sin(k) * t_m) / l)));
	}
	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):
	t_2 = k / math.cos(k)
	t_3 = math.sin(k) / l
	tmp = 0
	if t_m <= 4.6e-209:
		tmp = 2.0 / (((t_3 * t_2) * ((math.sin(k) * k) * t_m)) / l)
	else:
		tmp = 2.0 / ((t_2 * t_3) * (k * ((math.sin(k) * t_m) / l)))
	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(k / cos(k))
	t_3 = Float64(sin(k) / l)
	tmp = 0.0
	if (t_m <= 4.6e-209)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_3 * t_2) * Float64(Float64(sin(k) * k) * t_m)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 * t_3) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	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)
	t_2 = k / cos(k);
	t_3 = sin(k) / l;
	tmp = 0.0;
	if (t_m <= 4.6e-209)
		tmp = 2.0 / (((t_3 * t_2) * ((sin(k) * k) * t_m)) / l);
	else
		tmp = 2.0 / ((t_2 * t_3) * (k * ((sin(k) * t_m) / l)));
	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_] := Block[{t$95$2 = N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.6e-209], N[(2.0 / N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 4.5999999999999999e-209

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 94.6%

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

   11. Applied rewrites 93.7%

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

   if 4.5999999999999999e-209 < t

   1. Initial program 41.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 99.6%

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

Alternative 2: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t\_m}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t\_m}\right)}^{2}\right) - 1\right) \leq 10^{+120}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m \cdot t\_m}{\ell} \cdot \frac{t\_m}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\frac{k}{t\_m} \cdot \frac{k}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \]

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 (<=
       (*
        (* (* (/ (pow t_m 3.0) (* l l)) (sin k)) (tan k))
        (- (+ 1.0 (pow (/ k t_m) 2.0)) 1.0))
       1e+120)
    (/
     2.0
     (*
      (* (* (* (/ (* t_m t_m) l) (/ t_m l)) (sin k)) (tan k))
      (* (/ k t_m) (/ k t_m))))
    (/ 2.0 (* (* k (/ (sin k) l)) (* k (/ (* (sin k) 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) {
	double tmp;
	if (((((pow(t_m, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t_m), 2.0)) - 1.0)) <= 1e+120) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / ((k * (sin(k) / l)) * (k * ((sin(k) * t_m) / l)));
	}
	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 ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t_m) ** 2.0d0)) - 1.0d0)) <= 1d+120) then
        tmp = 2.0d0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * ((k / t_m) * (k / t_m)))
    else
        tmp = 2.0d0 / ((k * (sin(k) / l)) * (k * ((sin(k) * t_m) / l)))
    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 (((((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+120) {
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * Math.sin(k)) * Math.tan(k)) * ((k / t_m) * (k / t_m)));
	} else {
		tmp = 2.0 / ((k * (Math.sin(k) / l)) * (k * ((Math.sin(k) * t_m) / l)));
	}
	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 ((((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+120:
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * math.sin(k)) * math.tan(k)) * ((k / t_m) * (k / t_m)))
	else:
		tmp = 2.0 / ((k * (math.sin(k) / l)) * (k * ((math.sin(k) * t_m) / 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 (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+120)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(Float64(t_m * t_m) / l) * Float64(t_m / l)) * sin(k)) * tan(k)) * Float64(Float64(k / t_m) * Float64(k / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(sin(k) / l)) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	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 ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t_m) ^ 2.0)) - 1.0)) <= 1e+120)
		tmp = 2.0 / ((((((t_m * t_m) / l) * (t_m / l)) * sin(k)) * tan(k)) * ((k / t_m) * (k / t_m)));
	else
		tmp = 2.0 / ((k * (sin(k) / l)) * (k * ((sin(k) * t_m) / l)));
	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[(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], 1e+120], N[(2.0 / N[(N[(N[(N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / t$95$m), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t #s(literal 3 binary64)) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (-.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 k t) #s(literal 2 binary64))) #s(literal 1 binary64))) < 9.9999999999999998e119

   1. Initial program 93.0%

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

      1. lift-var N/A

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

      2. lift-literal N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{\mathsf{Rewrite<=}\left(lift-var-spec, t\right)}^{\color{blue}{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      3. lift-pow.f64 N/A

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

      4. lift-var N/A

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

      5. lift-var N/A

         \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\mathsf{Rewrite<=}\left(lift-var-spec, \ell\right) \cdot \color{blue}{\ell}} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-var N/A

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

      12. lift-var N/A

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

      13. lift-var N/A

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

      14. lower-/.f64 N/A

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

      15. lift-var N/A

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

      16. lift-var 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 95.8%

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

   if 9.9999999999999998e119 < (*.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 10.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 63.2%

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

   7. Applied rewrites 79.8%

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

   9. Applied rewrites 94.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 69.9%

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

Alternative 3: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\sin k}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{2}{\frac{\left(t\_2 \cdot k\right) \cdot \left(\left(\sin k \cdot k\right) \cdot t\_m\right)}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k} \cdot t\_2\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

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 (/ (sin k) l)))
   (*
    t_s
    (if (<= t_m 8.5e-216)
      (/ 2.0 (/ (* (* t_2 k) (* (* (sin k) k) t_m)) (* (cos k) l)))
      (/ 2.0 (* (* (/ k (cos k)) t_2) (* k (/ (* (sin k) 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) {
	double t_2 = sin(k) / l;
	double tmp;
	if (t_m <= 8.5e-216) {
		tmp = 2.0 / (((t_2 * k) * ((sin(k) * k) * t_m)) / (cos(k) * l));
	} else {
		tmp = 2.0 / (((k / cos(k)) * t_2) * (k * ((sin(k) * t_m) / l)));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = sin(k) / l
    if (t_m <= 8.5d-216) then
        tmp = 2.0d0 / (((t_2 * k) * ((sin(k) * k) * t_m)) / (cos(k) * l))
    else
        tmp = 2.0d0 / (((k / cos(k)) * t_2) * (k * ((sin(k) * t_m) / l)))
    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 t_2 = Math.sin(k) / l;
	double tmp;
	if (t_m <= 8.5e-216) {
		tmp = 2.0 / (((t_2 * k) * ((Math.sin(k) * k) * t_m)) / (Math.cos(k) * l));
	} else {
		tmp = 2.0 / (((k / Math.cos(k)) * t_2) * (k * ((Math.sin(k) * t_m) / l)));
	}
	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):
	t_2 = math.sin(k) / l
	tmp = 0
	if t_m <= 8.5e-216:
		tmp = 2.0 / (((t_2 * k) * ((math.sin(k) * k) * t_m)) / (math.cos(k) * l))
	else:
		tmp = 2.0 / (((k / math.cos(k)) * t_2) * (k * ((math.sin(k) * t_m) / l)))
	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(sin(k) / l)
	tmp = 0.0
	if (t_m <= 8.5e-216)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * k) * Float64(Float64(sin(k) * k) * t_m)) / Float64(cos(k) * l)));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / cos(k)) * t_2) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	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)
	t_2 = sin(k) / l;
	tmp = 0.0;
	if (t_m <= 8.5e-216)
		tmp = 2.0 / (((t_2 * k) * ((sin(k) * k) * t_m)) / (cos(k) * l));
	else
		tmp = 2.0 / (((k / cos(k)) * t_2) * (k * ((sin(k) * t_m) / l)));
	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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 8.5e-216], N[(2.0 / N[(N[(N[(t$95$2 * k), $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.50000000000000003e-216

   1. Initial program 36.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 94.6%

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

   11. Applied rewrites 93.7%

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

   if 8.50000000000000003e-216 < t

   1. Initial program 41.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 71.2%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 99.6%

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

Alternative 4: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sin k \cdot k\\ t_3 := \cos k \cdot \ell\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{\sin k}{\ell} \cdot k\right) \cdot \left(t\_2 \cdot t\_m\right)}{t\_3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t\_2}{t\_3} \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

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 (* (sin k) k)) (t_3 (* (cos k) l)))
   (*
    t_s
    (if (<= t_m 2e-202)
      (/ 2.0 (/ (* (* (/ (sin k) l) k) (* t_2 t_m)) t_3))
      (/ 2.0 (* (/ t_2 t_3) (* k (/ (* (sin k) 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) {
	double t_2 = sin(k) * k;
	double t_3 = cos(k) * l;
	double tmp;
	if (t_m <= 2e-202) {
		tmp = 2.0 / ((((sin(k) / l) * k) * (t_2 * t_m)) / t_3);
	} else {
		tmp = 2.0 / ((t_2 / t_3) * (k * ((sin(k) * t_m) / l)));
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = sin(k) * k
    t_3 = cos(k) * l
    if (t_m <= 2d-202) then
        tmp = 2.0d0 / ((((sin(k) / l) * k) * (t_2 * t_m)) / t_3)
    else
        tmp = 2.0d0 / ((t_2 / t_3) * (k * ((sin(k) * t_m) / l)))
    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 t_2 = Math.sin(k) * k;
	double t_3 = Math.cos(k) * l;
	double tmp;
	if (t_m <= 2e-202) {
		tmp = 2.0 / ((((Math.sin(k) / l) * k) * (t_2 * t_m)) / t_3);
	} else {
		tmp = 2.0 / ((t_2 / t_3) * (k * ((Math.sin(k) * t_m) / l)));
	}
	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):
	t_2 = math.sin(k) * k
	t_3 = math.cos(k) * l
	tmp = 0
	if t_m <= 2e-202:
		tmp = 2.0 / ((((math.sin(k) / l) * k) * (t_2 * t_m)) / t_3)
	else:
		tmp = 2.0 / ((t_2 / t_3) * (k * ((math.sin(k) * t_m) / l)))
	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(sin(k) * k)
	t_3 = Float64(cos(k) * l)
	tmp = 0.0
	if (t_m <= 2e-202)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) / l) * k) * Float64(t_2 * t_m)) / t_3));
	else
		tmp = Float64(2.0 / Float64(Float64(t_2 / t_3) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	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)
	t_2 = sin(k) * k;
	t_3 = cos(k) * l;
	tmp = 0.0;
	if (t_m <= 2e-202)
		tmp = 2.0 / ((((sin(k) / l) * k) * (t_2 * t_m)) / t_3);
	else
		tmp = 2.0 / ((t_2 / t_3) * (k * ((sin(k) * t_m) / l)));
	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_] := Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2e-202], N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(t$95$2 * t$95$m), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t$95$2 / t$95$3), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.0000000000000001e-202

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 85.3%

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

   9. Applied rewrites 94.7%

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

   11. Applied rewrites 93.8%

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

   if 2.0000000000000001e-202 < t

   1. Initial program 41.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 88.2%

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

   9. Applied rewrites 99.6%

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

   11. Applied rewrites 98.6%

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

Alternative 5: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \cos k \cdot \ell\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2} \cdot \frac{t\_m}{\ell}}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}{\ell \cdot t\_2}}\\ \end{array} \end{array} \end{array} \]

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 (* (cos k) l)))
   (*
    t_s
    (if (<= k 1.45e-37)
      (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) t_m) l))))
      (if (<= k 1.25e+151)
        (/ 2.0 (/ (* (pow (* k (sin k)) 2.0) (/ t_m l)) t_2))
        (/ 2.0 (* k (/ (* (* k t_m) (pow (sin k) 2.0)) (* l t_2)))))))))

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 = cos(k) * l;
	double tmp;
	if (k <= 1.45e-37) {
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	} else if (k <= 1.25e+151) {
		tmp = 2.0 / ((pow((k * sin(k)), 2.0) * (t_m / l)) / t_2);
	} else {
		tmp = 2.0 / (k * (((k * t_m) * pow(sin(k), 2.0)) / (l * t_2)));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = cos(k) * l
    if (k <= 1.45d-37) then
        tmp = 2.0d0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)))
    else if (k <= 1.25d+151) then
        tmp = 2.0d0 / ((((k * sin(k)) ** 2.0d0) * (t_m / l)) / t_2)
    else
        tmp = 2.0d0 / (k * (((k * t_m) * (sin(k) ** 2.0d0)) / (l * t_2)))
    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 t_2 = Math.cos(k) * l;
	double tmp;
	if (k <= 1.45e-37) {
		tmp = 2.0 / (((k / l) * k) * (k * ((Math.sin(k) * t_m) / l)));
	} else if (k <= 1.25e+151) {
		tmp = 2.0 / ((Math.pow((k * Math.sin(k)), 2.0) * (t_m / l)) / t_2);
	} else {
		tmp = 2.0 / (k * (((k * t_m) * Math.pow(Math.sin(k), 2.0)) / (l * t_2)));
	}
	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):
	t_2 = math.cos(k) * l
	tmp = 0
	if k <= 1.45e-37:
		tmp = 2.0 / (((k / l) * k) * (k * ((math.sin(k) * t_m) / l)))
	elif k <= 1.25e+151:
		tmp = 2.0 / ((math.pow((k * math.sin(k)), 2.0) * (t_m / l)) / t_2)
	else:
		tmp = 2.0 / (k * (((k * t_m) * math.pow(math.sin(k), 2.0)) / (l * t_2)))
	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(cos(k) * l)
	tmp = 0.0
	if (k <= 1.45e-37)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	elseif (k <= 1.25e+151)
		tmp = Float64(2.0 / Float64(Float64((Float64(k * sin(k)) ^ 2.0) * Float64(t_m / l)) / t_2));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k * t_m) * (sin(k) ^ 2.0)) / Float64(l * t_2))));
	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)
	t_2 = cos(k) * l;
	tmp = 0.0;
	if (k <= 1.45e-37)
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	elseif (k <= 1.25e+151)
		tmp = 2.0 / ((((k * sin(k)) ^ 2.0) * (t_m / l)) / t_2);
	else
		tmp = 2.0 / (k * (((k * t_m) * (sin(k) ^ 2.0)) / (l * t_2)));
	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_] := Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.45e-37], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.25e+151], N[(2.0 / N[(N[(N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.45000000000000002e-37

   1. Initial program 42.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 88.0%

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

   9. Applied rewrites 96.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.5%

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

   if 1.45000000000000002e-37 < k < 1.2500000000000001e151

   1. Initial program 30.4%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 88.8%

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

   7. Applied rewrites 95.2%

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

   9. Applied rewrites 95.2%

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

   if 1.2500000000000001e151 < k

   1. Initial program 26.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 59.7%

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

   9. Applied rewrites 73.2%

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

Alternative 6: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-39}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \left(\frac{t\_m}{\ell} \cdot k\right)\right) \cdot \frac{k}{\cos k \cdot \ell}}\\ \end{array} \end{array} \]

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 1e-39)
    (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) t_m) l))))
    (/ 2.0 (* (* (pow (sin k) 2.0) (* (/ t_m l) k)) (/ k (* (cos 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) {
	double tmp;
	if (k <= 1e-39) {
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) * ((t_m / l) * k)) * (k / (cos(k) * l)));
	}
	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 <= 1d-39) then
        tmp = 2.0d0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)))
    else
        tmp = 2.0d0 / (((sin(k) ** 2.0d0) * ((t_m / l) * k)) * (k / (cos(k) * l)))
    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 <= 1e-39) {
		tmp = 2.0 / (((k / l) * k) * (k * ((Math.sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) * ((t_m / l) * k)) * (k / (Math.cos(k) * l)));
	}
	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 <= 1e-39:
		tmp = 2.0 / (((k / l) * k) * (k * ((math.sin(k) * t_m) / l)))
	else:
		tmp = 2.0 / ((math.pow(math.sin(k), 2.0) * ((t_m / l) * k)) * (k / (math.cos(k) * 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 <= 1e-39)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) * Float64(Float64(t_m / l) * k)) * Float64(k / Float64(cos(k) * l))));
	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 <= 1e-39)
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	else
		tmp = 2.0 / (((sin(k) ^ 2.0) * ((t_m / l) * k)) * (k / (cos(k) * l)));
	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, 1e-39], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] * N[(k / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999929e-40

   1. Initial program 42.4%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 88.0%

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

   9. Applied rewrites 96.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.4%

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

   if 9.99999999999999929e-40 < k

   1. Initial program 28.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 82.1%

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

   9. Applied rewrites 96.1%

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

Alternative 7: 93.6% accurate, 1.3× speedup

Math

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

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
  (/ 2.0 (* (/ (* (sin k) k) (* (cos k) l)) (* k (/ (* (sin k) 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 * (2.0 / (((sin(k) * k) / (cos(k) * l)) * (k * ((sin(k) * 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 * (2.0d0 / (((sin(k) * k) / (cos(k) * l)) * (k * ((sin(k) * 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 * (2.0 / (((Math.sin(k) * k) / (Math.cos(k) * l)) * (k * ((Math.sin(k) * 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 * (2.0 / (((math.sin(k) * k) / (math.cos(k) * l)) * (k * ((math.sin(k) * 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(2.0 / Float64(Float64(Float64(sin(k) * k) / Float64(cos(k) * l)) * Float64(k * Float64(Float64(sin(k) * 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 * (2.0 / (((sin(k) * k) / (cos(k) * l)) * (k * ((sin(k) * 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[(2.0 / N[(N[(N[(N[Sin[k], $MachinePrecision] * k), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 74.2%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 96.6%

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

11. Applied rewrites 95.9%

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

Alternative 8: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k \cdot \frac{\sin k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot t\_2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \sin k\right)}^{2} \cdot \frac{t\_m}{\ell \cdot \left(\cos k \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* k (/ (* (sin k) t_m) l))))
   (*
    t_s
    (if (<= k 4.8e-26)
      (/ 2.0 (* (* (/ k l) k) t_2))
      (if (<= k 1.6e+149)
        (/ 2.0 (* (pow (* k (sin k)) 2.0) (/ t_m (* l (* (cos k) l)))))
        (/ 2.0 (* (* k (/ (sin k) l)) t_2)))))))

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 = k * ((sin(k) * t_m) / l);
	double tmp;
	if (k <= 4.8e-26) {
		tmp = 2.0 / (((k / l) * k) * t_2);
	} else if (k <= 1.6e+149) {
		tmp = 2.0 / (pow((k * sin(k)), 2.0) * (t_m / (l * (cos(k) * l))));
	} else {
		tmp = 2.0 / ((k * (sin(k) / l)) * t_2);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = k * ((sin(k) * t_m) / l)
    if (k <= 4.8d-26) then
        tmp = 2.0d0 / (((k / l) * k) * t_2)
    else if (k <= 1.6d+149) then
        tmp = 2.0d0 / (((k * sin(k)) ** 2.0d0) * (t_m / (l * (cos(k) * l))))
    else
        tmp = 2.0d0 / ((k * (sin(k) / l)) * t_2)
    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 t_2 = k * ((Math.sin(k) * t_m) / l);
	double tmp;
	if (k <= 4.8e-26) {
		tmp = 2.0 / (((k / l) * k) * t_2);
	} else if (k <= 1.6e+149) {
		tmp = 2.0 / (Math.pow((k * Math.sin(k)), 2.0) * (t_m / (l * (Math.cos(k) * l))));
	} else {
		tmp = 2.0 / ((k * (Math.sin(k) / l)) * t_2);
	}
	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):
	t_2 = k * ((math.sin(k) * t_m) / l)
	tmp = 0
	if k <= 4.8e-26:
		tmp = 2.0 / (((k / l) * k) * t_2)
	elif k <= 1.6e+149:
		tmp = 2.0 / (math.pow((k * math.sin(k)), 2.0) * (t_m / (l * (math.cos(k) * l))))
	else:
		tmp = 2.0 / ((k * (math.sin(k) / l)) * t_2)
	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(k * Float64(Float64(sin(k) * t_m) / l))
	tmp = 0.0
	if (k <= 4.8e-26)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * t_2));
	elseif (k <= 1.6e+149)
		tmp = Float64(2.0 / Float64((Float64(k * sin(k)) ^ 2.0) * Float64(t_m / Float64(l * Float64(cos(k) * l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(sin(k) / l)) * t_2));
	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)
	t_2 = k * ((sin(k) * t_m) / l);
	tmp = 0.0;
	if (k <= 4.8e-26)
		tmp = 2.0 / (((k / l) * k) * t_2);
	elseif (k <= 1.6e+149)
		tmp = 2.0 / (((k * sin(k)) ^ 2.0) * (t_m / (l * (cos(k) * l))));
	else
		tmp = 2.0 / ((k * (sin(k) / l)) * t_2);
	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_] := Block[{t$95$2 = N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 4.8e-26], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+149], N[(2.0 / N[(N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m / N[(l * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 4.8000000000000002e-26

   1. Initial program 42.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 88.2%

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

   9. Applied rewrites 96.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.7%

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

   if 4.8000000000000002e-26 < k < 1.6000000000000001e149

   1. Initial program 27.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 88.3%

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

   if 1.6000000000000001e149 < k

   1. Initial program 26.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 59.7%

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

   9. Applied rewrites 97.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 60.9%

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

Alternative 9: 82.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\left(k \cdot t\_m\right) \cdot {\sin k}^{2}}{\ell \cdot \left(\cos k \cdot \ell\right)}}\\ \end{array} \end{array} \]

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 3.8e-13)
    (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) t_m) l))))
    (/ 2.0 (* k (/ (* (* k t_m) (pow (sin k) 2.0)) (* l (* (cos 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) {
	double tmp;
	if (k <= 3.8e-13) {
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / (k * (((k * t_m) * pow(sin(k), 2.0)) / (l * (cos(k) * l))));
	}
	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 <= 3.8d-13) then
        tmp = 2.0d0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)))
    else
        tmp = 2.0d0 / (k * (((k * t_m) * (sin(k) ** 2.0d0)) / (l * (cos(k) * l))))
    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 <= 3.8e-13) {
		tmp = 2.0 / (((k / l) * k) * (k * ((Math.sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / (k * (((k * t_m) * Math.pow(Math.sin(k), 2.0)) / (l * (Math.cos(k) * l))));
	}
	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 <= 3.8e-13:
		tmp = 2.0 / (((k / l) * k) * (k * ((math.sin(k) * t_m) / l)))
	else:
		tmp = 2.0 / (k * (((k * t_m) * math.pow(math.sin(k), 2.0)) / (l * (math.cos(k) * 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 <= 3.8e-13)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k * t_m) * (sin(k) ^ 2.0)) / Float64(l * Float64(cos(k) * l)))));
	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 <= 3.8e-13)
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	else
		tmp = 2.0 / (k * (((k * t_m) * (sin(k) ^ 2.0)) / (l * (cos(k) * l))));
	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, 3.8e-13], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(k * t$95$m), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.8e-13

   1. Initial program 42.7%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 88.5%

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

   9. Applied rewrites 96.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.1%

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

   if 3.8e-13 < k

   1. Initial program 26.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 79.9%

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

   9. Applied rewrites 82.6%

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

Alternative 10: 80.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := k \cdot \frac{\sin k \cdot t\_m}{\ell}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot t\_2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{\cos k}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\_m\right)} \cdot \left(\ell \cdot \left(\ell + \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot t\_2}\\ \end{array} \end{array} \end{array} \]

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 (* k (/ (* (sin k) t_m) l))))
   (*
    t_s
    (if (<= k 1.9e-15)
      (/ 2.0 (* (* (/ k l) k) t_2))
      (if (<= k 1.6e+149)
        (* (/ (cos k) (* (* k k) (* (pow (sin k) 2.0) t_m))) (* l (+ l l)))
        (/ 2.0 (* (* k (/ (sin k) l)) t_2)))))))

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 = k * ((sin(k) * t_m) / l);
	double tmp;
	if (k <= 1.9e-15) {
		tmp = 2.0 / (((k / l) * k) * t_2);
	} else if (k <= 1.6e+149) {
		tmp = (cos(k) / ((k * k) * (pow(sin(k), 2.0) * t_m))) * (l * (l + l));
	} else {
		tmp = 2.0 / ((k * (sin(k) / l)) * t_2);
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = k * ((sin(k) * t_m) / l)
    if (k <= 1.9d-15) then
        tmp = 2.0d0 / (((k / l) * k) * t_2)
    else if (k <= 1.6d+149) then
        tmp = (cos(k) / ((k * k) * ((sin(k) ** 2.0d0) * t_m))) * (l * (l + l))
    else
        tmp = 2.0d0 / ((k * (sin(k) / l)) * t_2)
    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 t_2 = k * ((Math.sin(k) * t_m) / l);
	double tmp;
	if (k <= 1.9e-15) {
		tmp = 2.0 / (((k / l) * k) * t_2);
	} else if (k <= 1.6e+149) {
		tmp = (Math.cos(k) / ((k * k) * (Math.pow(Math.sin(k), 2.0) * t_m))) * (l * (l + l));
	} else {
		tmp = 2.0 / ((k * (Math.sin(k) / l)) * t_2);
	}
	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):
	t_2 = k * ((math.sin(k) * t_m) / l)
	tmp = 0
	if k <= 1.9e-15:
		tmp = 2.0 / (((k / l) * k) * t_2)
	elif k <= 1.6e+149:
		tmp = (math.cos(k) / ((k * k) * (math.pow(math.sin(k), 2.0) * t_m))) * (l * (l + l))
	else:
		tmp = 2.0 / ((k * (math.sin(k) / l)) * t_2)
	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(k * Float64(Float64(sin(k) * t_m) / l))
	tmp = 0.0
	if (k <= 1.9e-15)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * t_2));
	elseif (k <= 1.6e+149)
		tmp = Float64(Float64(cos(k) / Float64(Float64(k * k) * Float64((sin(k) ^ 2.0) * t_m))) * Float64(l * Float64(l + l)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(sin(k) / l)) * t_2));
	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)
	t_2 = k * ((sin(k) * t_m) / l);
	tmp = 0.0;
	if (k <= 1.9e-15)
		tmp = 2.0 / (((k / l) * k) * t_2);
	elseif (k <= 1.6e+149)
		tmp = (cos(k) / ((k * k) * ((sin(k) ^ 2.0) * t_m))) * (l * (l + l));
	else
		tmp = 2.0 / ((k * (sin(k) / l)) * t_2);
	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_] := Block[{t$95$2 = N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.9e-15], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+149], N[(N[(N[Cos[k], $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 1.9000000000000001e-15

   1. Initial program 42.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 88.4%

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

   9. Applied rewrites 96.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 82.0%

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

   if 1.9000000000000001e-15 < k < 1.6000000000000001e149

   1. Initial program 24.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.5%

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

   if 1.6000000000000001e149 < k

   1. Initial program 26.9%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 59.7%

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

   9. Applied rewrites 97.5%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 60.9%

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

Alternative 11: 78.0% accurate, 1.8× speedup

Math

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

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 (/ 2.0 (* (* k (/ (sin k) l)) (* k (/ (* (sin k) 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 * (2.0 / ((k * (sin(k) / l)) * (k * ((sin(k) * 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 * (2.0d0 / ((k * (sin(k) / l)) * (k * ((sin(k) * 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 * (2.0 / ((k * (Math.sin(k) / l)) * (k * ((Math.sin(k) * 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 * (2.0 / ((k * (math.sin(k) / l)) * (k * ((math.sin(k) * 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(2.0 / Float64(Float64(k * Float64(sin(k) / l)) * Float64(k * Float64(Float64(sin(k) * 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 * (2.0 / ((k * (sin(k) / l)) * (k * ((sin(k) * 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[(2.0 / N[(N[(k * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 74.2%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 96.6%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 77.6%

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

Alternative 12: 77.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot k\right) \cdot \frac{k \cdot k}{\cos k}}\\ \end{array} \end{array} \]

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 6.8e-28)
    (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) t_m) l))))
    (/ 2.0 (* (* (* (/ (/ t_m l) l) k) k) (/ (* k k) (cos 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 <= 6.8e-28) {
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / (((((t_m / l) / l) * k) * k) * ((k * k) / cos(k)));
	}
	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 <= 6.8d-28) then
        tmp = 2.0d0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)))
    else
        tmp = 2.0d0 / (((((t_m / l) / l) * k) * k) * ((k * k) / cos(k)))
    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 <= 6.8e-28) {
		tmp = 2.0 / (((k / l) * k) * (k * ((Math.sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / (((((t_m / l) / l) * k) * k) * ((k * k) / Math.cos(k)));
	}
	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 <= 6.8e-28:
		tmp = 2.0 / (((k / l) * k) * (k * ((math.sin(k) * t_m) / l)))
	else:
		tmp = 2.0 / (((((t_m / l) / l) * k) * k) * ((k * k) / math.cos(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 <= 6.8e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) / l) * k) * k) * Float64(Float64(k * k) / cos(k))));
	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 <= 6.8e-28)
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	else
		tmp = 2.0 / (((((t_m / l) / l) * k) * k) * ((k * k) / cos(k)));
	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, 6.8e-28], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 6.8000000000000001e-28

   1. Initial program 42.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 88.2%

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

   9. Applied rewrites 96.8%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.7%

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

   if 6.8000000000000001e-28 < k

   1. Initial program 27.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 64.5%

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

Alternative 13: 77.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(k \cdot \frac{\sin k \cdot t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{t\_m}{\ell} \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}{\ell \cdot \cos k}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.45e-37)
    (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) t_m) l))))
    (/ 2.0 (/ (* (* (/ t_m l) (* k k)) (* k k)) (* l (cos 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 <= 1.45e-37) {
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / ((((t_m / l) * (k * k)) * (k * k)) / (l * cos(k)));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, t_m, l, k)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.45d-37) then
        tmp = 2.0d0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)))
    else
        tmp = 2.0d0 / ((((t_m / l) * (k * k)) * (k * k)) / (l * cos(k)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.45e-37) {
		tmp = 2.0 / (((k / l) * k) * (k * ((Math.sin(k) * t_m) / l)));
	} else {
		tmp = 2.0 / ((((t_m / l) * (k * k)) * (k * k)) / (l * Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.45e-37:
		tmp = 2.0 / (((k / l) * k) * (k * ((math.sin(k) * t_m) / l)))
	else:
		tmp = 2.0 / ((((t_m / l) * (k * k)) * (k * k)) / (l * math.cos(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 <= 1.45e-37)
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * t_m) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t_m / l) * Float64(k * k)) * Float64(k * k)) / Float64(l * cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.45e-37)
		tmp = 2.0 / (((k / l) * k) * (k * ((sin(k) * t_m) / l)));
	else
		tmp = 2.0 / ((((t_m / l) * (k * k)) * (k * k)) / (l * cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.45e-37], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.45000000000000002e-37

   1. Initial program 42.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 73.5%

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

   7. Applied rewrites 88.0%

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

   9. Applied rewrites 96.7%

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

   10. Taylor expanded in k around 0

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

   12. Applied rewrites 81.5%

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

   if 1.45000000000000002e-37 < k

   1. Initial program 29.1%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 64.4%

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

Alternative 14: 76.5% accurate, 3.0× speedup

Math

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

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 (/ 2.0 (* (* (/ k l) k) (* k (/ (* (sin k) 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 * (2.0 / (((k / l) * k) * (k * ((sin(k) * 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 * (2.0d0 / (((k / l) * k) * (k * ((sin(k) * 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 * (2.0 / (((k / l) * k) * (k * ((Math.sin(k) * 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 * (2.0 / (((k / l) * k) * (k * ((math.sin(k) * 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(2.0 / Float64(Float64(Float64(k / l) * k) * Float64(k * Float64(Float64(sin(k) * 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 * (2.0 / (((k / l) * k) * (k * ((sin(k) * 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[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * N[(N[(N[Sin[k], $MachinePrecision] * t$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 74.2%

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

7. Applied rewrites 86.4%

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

9. Applied rewrites 96.6%

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

10. Taylor expanded in k around 0

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

12. Applied rewrites 75.9%

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

Alternative 15: 73.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot k}}{t\_m}\\ \end{array} \end{array} \]

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 2e+41)
    (/ 2.0 (* (* (* (* (/ t_m l) k) (/ k l)) k) k))
    (/ (* (/ (+ l l) k) (/ l (* (* k 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 (t_m <= 2e+41) {
		tmp = 2.0 / (((((t_m / l) * k) * (k / l)) * k) * k);
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2d+41) then
        tmp = 2.0d0 / (((((t_m / l) * k) * (k / l)) * k) * k)
    else
        tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2e+41) {
		tmp = 2.0 / (((((t_m / l) * k) * (k / l)) * k) * k);
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 t_m <= 2e+41:
		tmp = 2.0 / (((((t_m / l) * k) * (k / l)) * k) * k)
	else:
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2e+41)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) * k) * Float64(k / l)) * k) * k));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(Float64(k * 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 (t_m <= 2e+41)
		tmp = 2.0 / (((((t_m / l) * k) * (k / l)) * k) * k);
	else
		tmp = (((l + l) / k) * (l / ((k * 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[t$95$m, 2e+41], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000001e41

   1. Initial program 41.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

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

   12. Applied rewrites 71.7%

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

   if 2.00000000000000001e41 < t

   1. Initial program 27.5%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in l around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot k\right) \cdot \frac{k}{\ell}\right) \cdot k\right) \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot k}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(\frac{t\_m}{\ell} \cdot k\right) \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot k}}{t\_m}\\ \end{array} \end{array} \]

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 3.6e+90)
    (/ 2.0 (* (/ (* k k) l) (* (* (/ t_m l) k) k)))
    (/ (* (/ (+ l l) k) (/ l (* (* k 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 (t_m <= 3.6e+90) {
		tmp = 2.0 / (((k * k) / l) * (((t_m / l) * k) * k));
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 3.6d+90) then
        tmp = 2.0d0 / (((k * k) / l) * (((t_m / l) * k) * k))
    else
        tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 3.6e+90) {
		tmp = 2.0 / (((k * k) / l) * (((t_m / l) * k) * k));
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 t_m <= 3.6e+90:
		tmp = 2.0 / (((k * k) / l) * (((t_m / l) * k) * k))
	else:
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 3.6e+90)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(Float64(t_m / l) * k) * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(Float64(k * 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 (t_m <= 3.6e+90)
		tmp = 2.0 / (((k * k) / l) * (((t_m / l) * k) * k));
	else
		tmp = (((l + l) / k) * (l / ((k * 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[t$95$m, 3.6e+90], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(t$95$m / l), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.6e90

   1. Initial program 42.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.7%

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

   12. Applied rewrites 72.1%

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

   if 3.6e90 < t

   1. Initial program 18.9%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in l around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.4%

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

Alternative 17: 73.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\left(\frac{t\_m}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot k}}{t\_m}\\ \end{array} \end{array} \]

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 2e+41)
    (/ 2.0 (* (* (/ t_m l) (/ (* k k) l)) (* k k)))
    (/ (* (/ (+ l l) k) (/ l (* (* k 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 (t_m <= 2e+41) {
		tmp = 2.0 / (((t_m / l) * ((k * k) / l)) * (k * k));
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2d+41) then
        tmp = 2.0d0 / (((t_m / l) * ((k * k) / l)) * (k * k))
    else
        tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2e+41) {
		tmp = 2.0 / (((t_m / l) * ((k * k) / l)) * (k * k));
	} else {
		tmp = (((l + l) / k) * (l / ((k * 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 t_m <= 2e+41:
		tmp = 2.0 / (((t_m / l) * ((k * k) / l)) * (k * k))
	else:
		tmp = (((l + l) / k) * (l / ((k * 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 (t_m <= 2e+41)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_m / l) * Float64(Float64(k * k) / l)) * Float64(k * k)));
	else
		tmp = Float64(Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(Float64(k * 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 (t_m <= 2e+41)
		tmp = 2.0 / (((t_m / l) * ((k * k) / l)) * (k * k));
	else
		tmp = (((l + l) / k) * (l / ((k * 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[t$95$m, 2e+41], N[(2.0 / N[(N[(N[(t$95$m / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.00000000000000001e41

   1. Initial program 41.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.4%

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.2%

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

   if 2.00000000000000001e41 < t

   1. Initial program 27.5%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in l around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.9%

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

Alternative 18: 72.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell + \ell}{k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot k}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{\frac{t\_m}{\ell}}{\ell} \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot k}\\ \end{array} \end{array} \]

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 3e-40)
    (/ (* (/ (+ l l) k) (/ l (* (* k k) k))) t_m)
    (/ 2.0 (* (* (* (/ (/ t_m l) l) k) (* k 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 (l <= 3e-40) {
		tmp = (((l + l) / k) * (l / ((k * k) * k))) / t_m;
	} else {
		tmp = 2.0 / (((((t_m / l) / l) * k) * (k * k)) * k);
	}
	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 <= 3d-40) then
        tmp = (((l + l) / k) * (l / ((k * k) * k))) / t_m
    else
        tmp = 2.0d0 / (((((t_m / l) / l) * k) * (k * k)) * k)
    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 <= 3e-40) {
		tmp = (((l + l) / k) * (l / ((k * k) * k))) / t_m;
	} else {
		tmp = 2.0 / (((((t_m / l) / l) * k) * (k * k)) * k);
	}
	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 <= 3e-40:
		tmp = (((l + l) / k) * (l / ((k * k) * k))) / t_m
	else:
		tmp = 2.0 / (((((t_m / l) / l) * k) * (k * 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 (l <= 3e-40)
		tmp = Float64(Float64(Float64(Float64(l + l) / k) * Float64(l / Float64(Float64(k * k) * k))) / t_m);
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t_m / l) / l) * k) * Float64(k * k)) * k));
	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 <= 3e-40)
		tmp = (((l + l) / k) * (l / ((k * k) * k))) / t_m;
	else
		tmp = 2.0 / (((((t_m / l) / l) * k) * (k * k)) * k);
	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, 3e-40], N[(N[(N[(N[(l + l), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(N[(t$95$m / l), $MachinePrecision] / l), $MachinePrecision] * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.0000000000000002e-40

   1. Initial program 39.3%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in l around 0

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.6%

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

   if 3.0000000000000002e-40 < l

   1. Initial program 37.4%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 56.7%

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

   10. Applied rewrites 60.4%

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

Alternative 19: 64.2% accurate, 8.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t\_m \cdot \frac{k \cdot k}{\ell \cdot \ell}\right) \cdot \left(k \cdot k\right)}\\ \end{array} \end{array} \]

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 5.5e-167)
    (/ (/ 2.0 (* (* k k) (* k k))) t_m)
    (/ 2.0 (* (* t_m (/ (* k k) (* l l))) (* 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 <= 5.5e-167) {
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	} else {
		tmp = 2.0 / ((t_m * ((k * k) / (l * l))) * (k * k));
	}
	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 <= 5.5d-167) then
        tmp = (2.0d0 / ((k * k) * (k * k))) / t_m
    else
        tmp = 2.0d0 / ((t_m * ((k * k) / (l * l))) * (k * k))
    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 <= 5.5e-167) {
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	} else {
		tmp = 2.0 / ((t_m * ((k * k) / (l * l))) * (k * k));
	}
	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 <= 5.5e-167:
		tmp = (2.0 / ((k * k) * (k * k))) / t_m
	else:
		tmp = 2.0 / ((t_m * ((k * k) / (l * l))) * (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 <= 5.5e-167)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) / t_m);
	else
		tmp = Float64(2.0 / Float64(Float64(t_m * Float64(Float64(k * k) / Float64(l * l))) * Float64(k * k)));
	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 <= 5.5e-167)
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	else
		tmp = 2.0 / ((t_m * ((k * k) / (l * l))) * (k * k));
	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, 5.5e-167], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(2.0 / N[(N[(t$95$m * N[(N[(k * k), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 5.5000000000000003e-167

   1. Initial program 40.4%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 64.9%

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t} \]

   if 5.5000000000000003e-167 < k

   1. Initial program 35.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 86.2%

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

   8. Taylor expanded in k around 0

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

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

   12. Applied rewrites 64.3%

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

Alternative 20: 63.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell + \ell\right)}{\left(\left(k \cdot k\right) \cdot k\right) \cdot k}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.8e-86)
    (/ (/ 2.0 (* (* k k) (* k k))) t_m)
    (/ (/ (* l (+ l l)) (* (* (* k k) 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 <= 1.8e-86) {
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	} else {
		tmp = ((l * (l + l)) / (((k * k) * 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 (k <= 1.8d-86) then
        tmp = (2.0d0 / ((k * k) * (k * k))) / t_m
    else
        tmp = ((l * (l + l)) / (((k * k) * 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 (k <= 1.8e-86) {
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	} else {
		tmp = ((l * (l + l)) / (((k * k) * 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 k <= 1.8e-86:
		tmp = (2.0 / ((k * k) * (k * k))) / t_m
	else:
		tmp = ((l * (l + l)) / (((k * k) * 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 <= 1.8e-86)
		tmp = Float64(Float64(2.0 / Float64(Float64(k * k) * Float64(k * k))) / t_m);
	else
		tmp = Float64(Float64(Float64(l * Float64(l + l)) / Float64(Float64(Float64(k * k) * 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 (k <= 1.8e-86)
		tmp = (2.0 / ((k * k) * (k * k))) / t_m;
	else
		tmp = ((l * (l + l)) / (((k * k) * 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[k, 1.8e-86], N[(N[(2.0 / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.79999999999999983e-86

   1. Initial program 42.2%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 65.9%

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t} \]

   if 1.79999999999999983e-86 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 55.2%

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

Alternative 21: 63.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(k \cdot k\right) \cdot \left(k \cdot k\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{2}{t\_2}}{t\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell + \ell\right)}{t\_2}}{t\_m}\\ \end{array} \end{array} \end{array} \]

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 (* (* k k) (* k k))))
   (*
    t_s
    (if (<= k 1.8e-86) (/ (/ 2.0 t_2) t_m) (/ (/ (* l (+ l l)) t_2) 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 t_2 = (k * k) * (k * k);
	double tmp;
	if (k <= 1.8e-86) {
		tmp = (2.0 / t_2) / t_m;
	} else {
		tmp = ((l * (l + l)) / t_2) / 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) :: t_2
    real(8) :: tmp
    t_2 = (k * k) * (k * k)
    if (k <= 1.8d-86) then
        tmp = (2.0d0 / t_2) / t_m
    else
        tmp = ((l * (l + l)) / t_2) / 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 t_2 = (k * k) * (k * k);
	double tmp;
	if (k <= 1.8e-86) {
		tmp = (2.0 / t_2) / t_m;
	} else {
		tmp = ((l * (l + l)) / t_2) / 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):
	t_2 = (k * k) * (k * k)
	tmp = 0
	if k <= 1.8e-86:
		tmp = (2.0 / t_2) / t_m
	else:
		tmp = ((l * (l + l)) / t_2) / 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)
	t_2 = Float64(Float64(k * k) * Float64(k * k))
	tmp = 0.0
	if (k <= 1.8e-86)
		tmp = Float64(Float64(2.0 / t_2) / t_m);
	else
		tmp = Float64(Float64(Float64(l * Float64(l + l)) / t_2) / 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)
	t_2 = (k * k) * (k * k);
	tmp = 0.0;
	if (k <= 1.8e-86)
		tmp = (2.0 / t_2) / t_m;
	else
		tmp = ((l * (l + l)) / t_2) / 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_] := Block[{t$95$2 = N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 1.8e-86], N[(N[(2.0 / t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision], N[(N[(N[(l * N[(l + l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.79999999999999983e-86

   1. Initial program 42.2%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 65.9%

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

   7. Applied rewrites 65.9%

      \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t} \]

   if 1.79999999999999983e-86 < k

   1. Initial program 30.4%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 55.2%

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

Alternative 22: 71.5% accurate, 8.9× speedup

Math

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

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 l) k) (/ l (* (* k 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) {
	return t_s * ((((l + l) / k) * (l / ((k * k) * k))) / t_m);
}

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

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

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 62.7%

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

6. Taylor expanded in l around 0

   \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{4}}}{t} \]
7. Step-by-step derivation

8. Applied rewrites 71.8%

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

Alternative 23: 70.2% accurate, 8.9× speedup

Math

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

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 t_m) (/ (/ (+ l l) k) (* (* k 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) {
	return t_s * ((l / t_m) * (((l + l) / k) / ((k * k) * k)));
}

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

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

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 62.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 71.0%

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

Alternative 24: 44.6% accurate, 11.3× speedup

Math

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

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 l) (* (* k k) (* 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) {
	return t_s * (((l + l) / ((k * k) * (k * k))) / t_m);
}

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

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

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 62.7%

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

7. Applied rewrites 44.7%

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

Alternative 25: 57.7% accurate, 12.2× speedup

Math

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

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 (/ (/ 2.0 (* (* k k) (* 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) {
	return t_s * ((2.0 / ((k * k) * (k * k))) / t_m);
}

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 * ((2.0d0 / ((k * k) * (k * k))) / t_m)
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 * ((2.0 / ((k * k) * (k * k))) / t_m);
}

Python

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

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

Derivation

1. Initial program 38.7%

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

3. Taylor expanded in k around 0

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

5. Applied rewrites 62.7%

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

7. Applied rewrites 57.6%

   \[\leadsto \frac{\frac{2}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(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))))