Toniolo and Linder, Equation (10-)

Average Accuracy: 25.1% → 99.3%

Time: 33.1s

Precision: binary64

Cost: 20416

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{\sin k}\\ 2 \cdot \left(\frac{t_1}{t} \cdot \frac{t_1}{\frac{1}{\cos k}}\right) \end{array} \]

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 (/ (/ l k) (sin k))))
   (* 2.0 (* (/ t_1 t) (/ t_1 (/ 1.0 (cos k)))))))

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 = (l / k) / sin(k);
	return 2.0 * ((t_1 / t) * (t_1 / (1.0 / cos(k))));
}

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
    t_1 = (l / k) / sin(k)
    code = 2.0d0 * ((t_1 / t) * (t_1 / (1.0d0 / cos(k))))
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 = (l / k) / Math.sin(k);
	return 2.0 * ((t_1 / t) * (t_1 / (1.0 / Math.cos(k))));
}

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 = (l / k) / math.sin(k)
	return 2.0 * ((t_1 / t) * (t_1 / (1.0 / math.cos(k))))

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(Float64(l / k) / sin(k))
	return Float64(2.0 * Float64(Float64(t_1 / t) * Float64(t_1 / Float64(1.0 / cos(k)))))
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 = code(t, l, k)
	t_1 = (l / k) / sin(k);
	tmp = 2.0 * ((t_1 / t) * (t_1 / (1.0 / cos(k))));
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[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(t$95$1 / t), $MachinePrecision] * N[(t$95$1 / N[(1.0 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.1%

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

2. Simplified 36.9%

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

   [Start] 25.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   +-rgt-identity [<=] 25.1	\[ \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]
   associate-*l* [=>] 25.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]
   mul0-rgt [<=] 9.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]
   distribute-lft-in [<=] 25.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]
   +-rgt-identity [=>] 25.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   sub-neg [=>] 25.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}\right)} \]
   +-commutative [=>] 25.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + \left(-1\right)\right)\right)} \]
   associate-+l+ [=>] 36.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}\right)} \]
   metadata-eval [=>] 36.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{-1}\right)\right)\right)} \]
   metadata-eval [=>] 36.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
   +-rgt-identity [=>] 36.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

3. Taylor expanded in t around 0 65.0%

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

4. Simplified 63.1%

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

   [Start] 65.0	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
   unpow2 [=>] 65.0	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
   *-commutative [=>] 65.0	\[ 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
   associate-*r* [=>] 63.1	\[ 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}} \]
   times-frac [=>] 63.1	\[ 2 \cdot \color{blue}{\left(\frac{\ell \cdot \ell}{{k}^{2} \cdot {\sin k}^{2}} \cdot \frac{\cos k}{t}\right)} \]
   unpow2 [=>] 63.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {\sin k}^{2}} \cdot \frac{\cos k}{t}\right) \]
   associate-*l* [=>] 63.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {\sin k}^{2}\right)}} \cdot \frac{\cos k}{t}\right) \]

5. Taylor expanded in l around 0 63.1%

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

6. Simplified 87.8%

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

   [Start] 63.1	\[ 2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2} \cdot {\sin k}^{2}} \cdot \frac{\cos k}{t}\right) \]
   unpow2 [=>] 63.1	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {\sin k}^{2}} \cdot \frac{\cos k}{t}\right) \]
   unpow2 [=>] 63.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {\sin k}^{2}} \cdot \frac{\cos k}{t}\right) \]
   unpow2 [=>] 63.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \color{blue}{\left(\sin k \cdot \sin k\right)}} \cdot \frac{\cos k}{t}\right) \]
   swap-sqr [<=] 63.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}} \cdot \frac{\cos k}{t}\right) \]
   times-frac [=>] 87.8	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{k \cdot \sin k}\right)} \cdot \frac{\cos k}{t}\right) \]
   unpow2 [<=] 87.8	\[ 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}} \cdot \frac{\cos k}{t}\right) \]

7. Applied egg-rr 89.3%

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

   [Start] 87.8	\[ 2 \cdot \left({\left(\frac{\ell}{k \cdot \sin k}\right)}^{2} \cdot \frac{\cos k}{t}\right) \]
   clear-num [=>] 87.8	\[ 2 \cdot \left({\left(\frac{\ell}{k \cdot \sin k}\right)}^{2} \cdot \color{blue}{\frac{1}{\frac{t}{\cos k}}}\right) \]
   un-div-inv [=>] 87.9	\[ 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}} \]
   associate-/r* [=>] 89.3	\[ 2 \cdot \frac{{\color{blue}{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}}^{2}}{\frac{t}{\cos k}} \]

8. Applied egg-rr 99.3%

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

   [Start] 89.3	\[ 2 \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{\frac{t}{\cos k}} \]
   unpow2 [=>] 89.3	\[ 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}}}{\frac{t}{\cos k}} \]
   div-inv [=>] 89.3	\[ 2 \cdot \frac{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}}{\color{blue}{t \cdot \frac{1}{\cos k}}} \]
   times-frac [=>] 99.3	\[ 2 \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{k}}{\sin k}}{t} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{1}{\cos k}}\right)} \]

9. Final simplification 99.3%

   \[\leadsto 2 \cdot \left(\frac{\frac{\frac{\ell}{k}}{\sin k}}{t} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{\frac{1}{\cos k}}\right) \]

Alternatives

Alternative 1
Accuracy	93.0%
Cost	20624

\[\begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+56}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot {\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-211}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \leq 150000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{1}{{\left(\frac{\sin k}{\frac{\ell}{k}}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2
Accuracy	93.0%
Cost	20496

\[\begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ t_2 := 2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t}\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.8 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{-211}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \leq 150000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	93.1%
Cost	20496

\[\begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ t_2 := 2 \cdot \left(\frac{\cos k}{t} \cdot {\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}\right)\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{-212}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \leq 102000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	20288

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

Alternative 5
Accuracy	79.5%
Cost	14600

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \frac{\frac{\ell}{k} \cdot \frac{-0.3333333333333333}{t}}{\frac{k}{\ell}}\right)\\ \end{array} \]

Alternative 6
Accuracy	79.4%
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 7
Accuracy	79.8%
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1}}{k}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 8
Accuracy	63.8%
Cost	7424

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

Alternative 9
Accuracy	63.9%
Cost	960

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

Alternative 10
Accuracy	49.5%
Cost	832

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

Reproduce

herbie shell --seed 2023146 
(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))))