Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 0.5

Time: 29.7s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{-110} \lor \neg \left(k \leq 1.9 \cdot 10^{-74}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* k (/ k l))))
   (if (or (<= k -2.7e-110) (not (<= k 1.9e-74)))
     (* 2.0 (/ (* (/ (cos k) k) l) (* t (* (pow (sin k) 2.0) (/ k l)))))
     (/ 2.0 (* t_1 (* t t_1))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((k <= -2.7e-110) || !(k <= 1.9e-74)) {
		tmp = 2.0 * (((cos(k) / k) * l) / (t * (pow(sin(k), 2.0) * (k / l))));
	} else {
		tmp = 2.0 / (t_1 * (t * t_1));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = k * (k / l)
    if ((k <= (-2.7d-110)) .or. (.not. (k <= 1.9d-74))) then
        tmp = 2.0d0 * (((cos(k) / k) * l) / (t * ((sin(k) ** 2.0d0) * (k / l))))
    else
        tmp = 2.0d0 / (t_1 * (t * t_1))
    end if
    code = tmp
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));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = k * (k / l);
	double tmp;
	if ((k <= -2.7e-110) || !(k <= 1.9e-74)) {
		tmp = 2.0 * (((Math.cos(k) / k) * l) / (t * (Math.pow(Math.sin(k), 2.0) * (k / l))));
	} else {
		tmp = 2.0 / (t_1 * (t * t_1));
	}
	return tmp;
}

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))

↓

def code(t, l, k):
	t_1 = k * (k / l)
	tmp = 0
	if (k <= -2.7e-110) or not (k <= 1.9e-74):
		tmp = 2.0 * (((math.cos(k) / k) * l) / (t * (math.pow(math.sin(k), 2.0) * (k / l))))
	else:
		tmp = 2.0 / (t_1 * (t * t_1))
	return tmp

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

↓

function code(t, l, k)
	t_1 = Float64(k * Float64(k / l))
	tmp = 0.0
	if ((k <= -2.7e-110) || !(k <= 1.9e-74))
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) * l) / Float64(t * Float64((sin(k) ^ 2.0) * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * t_1)));
	end
	return tmp
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

↓

function tmp_2 = code(t, l, k)
	t_1 = k * (k / l);
	tmp = 0.0;
	if ((k <= -2.7e-110) || ~((k <= 1.9e-74)))
		tmp = 2.0 * (((cos(k) / k) * l) / (t * ((sin(k) ^ 2.0) * (k / l))));
	else
		tmp = 2.0 / (t_1 * (t * t_1));
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -2.7e-110], N[Not[LessEqual[k, 1.9e-74]], $MachinePrecision]], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * l), $MachinePrecision] / N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -2.6999999999999998e-110 or 1.8999999999999998e-74 < k

   1. Initial program 46.5

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

   2. Simplified 38.1

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 46.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 46.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 46.4	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 46.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 38.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 38.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.6

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

   4. Simplified 15.1

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

      [Start] 19.6	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 19.4	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 19.3	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 19.3	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]
      associate-/l* [=>] 15.1	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{{\sin k}^{2} \cdot t}{\ell}}}\right) \]
      *-commutative [=>] 15.1	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {\sin k}^{2}}}{\ell}}\right) \]

   5. Applied egg-rr 4.2

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

   6. Applied egg-rr 0.9

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

   7. Taylor expanded in k around inf 4.8

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

   8. Simplified 0.5

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

      [Start] 4.8	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{k \cdot \left({\sin k}^{2} \cdot t\right)}{\ell}} \]
      associate-*l/ [<=] 0.9	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\color{blue}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
      associate-*r* [=>] 0.5	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\color{blue}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot t}} \]
      *-commutative [<=] 0.5	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\color{blue}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right)} \cdot t} \]
      *-commutative [<=] 0.5	\[ 2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}} \]

   if -2.6999999999999998e-110 < k < 1.8999999999999998e-74

   1. Initial program 63.8

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

   2. Simplified 54.2

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

      [Start] 63.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 63.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 63.8	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 64.0	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 63.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 63.5	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 63.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 54.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 54.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 54.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 62.8

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

   4. Simplified 62.1

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

      [Start] 62.8	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      associate-*r/ [=>] 62.8	\[ \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]
      *-commutative [=>] 62.8	\[ \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 62.1	\[ \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
      unpow2 [=>] 62.1	\[ \frac{2}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}} \]

   5. Applied egg-rr 26.4

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

   6. Applied egg-rr 0.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{-110} \lor \neg \left(k \leq 1.9 \cdot 10^{-74}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.9
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -0.15 \lor \neg \left(k \leq 0.000104\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{k \cdot \left(\left(0.5 - \frac{\cos \left(k + k\right)}{2}\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -0.15 \lor \neg \left(k \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	3.9
Cost	14408

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\cos k}{k} \cdot \ell\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -0.15:\\ \;\;\;\;2 \cdot \frac{t_2}{k \cdot \left(t \cdot \frac{1 - t_1}{2 \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{k \cdot \left(\left(0.5 - \frac{t_1}{2}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	13.6
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.45 \cdot 10^{-65} \lor \neg \left(k \leq 1.95 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	26.1
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \frac{2}{t} \cdot \left(t_1 \cdot t_1\right) \end{array} \]

Alternative 6
Error	25.6
Cost	960

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

Alternative 7
Error	22.9
Cost	960

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \frac{2}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Reproduce

herbie shell --seed 2023032 
(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))))