Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 0.4

Time: 30.3s

Precision: binary64

Cost: 13760

Math

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

↓

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

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
 (* 2.0 (* (/ (/ (/ l k) (sin k)) t) (/ (/ l k) (tan 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) {
	return 2.0 * ((((l / k) / sin(k)) / t) * ((l / k) / tan(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
    code = 2.0d0 * ((((l / k) / sin(k)) / t) * ((l / k) / tan(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) {
	return 2.0 * ((((l / k) / Math.sin(k)) / t) * ((l / k) / Math.tan(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):
	return 2.0 * ((((l / k) / math.sin(k)) / t) * ((l / k) / math.tan(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)
	return Float64(2.0 * Float64(Float64(Float64(Float64(l / k) / sin(k)) / t) * Float64(Float64(l / k) / tan(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)
	tmp = 2.0 * ((((l / k) / sin(k)) / t) * ((l / k) / tan(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_] := N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3

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

2. Simplified 39.7

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

   [Start] 47.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 47.3	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 47.3	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 47.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 39.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 39.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 21.9

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

4. Applied egg-rr 21.8

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

5. Simplified 1.8

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

   [Start] 21.8	\[ \frac{2}{\tan k \cdot \left(\frac{k \cdot k}{\ell \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)} \]
   times-frac [=>] 7.4	\[ \frac{2}{\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{-\ell}\right)} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)} \]
   associate-*l* [=>] 1.8	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(\frac{k}{-\ell} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)\right)}} \]

6. Applied egg-rr 4.8

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

7. Simplified 0.4

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

   [Start] 4.8	\[ \frac{2}{\tan k} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)} \]
   associate-*r/ [=>] 4.5	\[ \color{blue}{\frac{\frac{2}{\tan k} \cdot \ell}{k \cdot \left(\sin k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}} \]
   associate-*r* [=>] 5.8	\[ \frac{\frac{2}{\tan k} \cdot \ell}{\color{blue}{\left(k \cdot \sin k\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}} \]
   associate-*r* [=>] 7.6	\[ \frac{\frac{2}{\tan k} \cdot \ell}{\color{blue}{\left(\left(k \cdot \sin k\right) \cdot t\right) \cdot \frac{k}{\ell}}} \]
   associate-/r* [=>] 6.5	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k} \cdot \ell}{\left(k \cdot \sin k\right) \cdot t}}{\frac{k}{\ell}}} \]
   associate-*r/ [<=] 6.5	\[ \frac{\color{blue}{\frac{2}{\tan k} \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}}}{\frac{k}{\ell}} \]
   associate-/l* [<=] 10.5	\[ \color{blue}{\frac{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \ell}{k}} \]
   associate-*r/ [<=] 6.5	\[ \color{blue}{\left(\frac{2}{\tan k} \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \frac{\ell}{k}} \]
   associate-*l/ [=>] 6.5	\[ \color{blue}{\frac{2 \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}}{\tan k}} \cdot \frac{\ell}{k} \]
   associate-*l/ [=>] 7.0	\[ \color{blue}{\frac{\left(2 \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \frac{\ell}{k}}{\tan k}} \]
   *-lft-identity [<=] 7.0	\[ \frac{\left(2 \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \color{blue}{\left(1 \cdot \frac{\ell}{k}\right)}}{\tan k} \]
   associate-*r/ [<=] 6.5	\[ \color{blue}{\left(2 \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \frac{1 \cdot \frac{\ell}{k}}{\tan k}} \]
   associate-*l/ [<=] 6.5	\[ \left(2 \cdot \frac{\ell}{\left(k \cdot \sin k\right) \cdot t}\right) \cdot \color{blue}{\left(\frac{1}{\tan k} \cdot \frac{\ell}{k}\right)} \]
   associate-*l* [=>] 6.5	\[ \color{blue}{2 \cdot \left(\frac{\ell}{\left(k \cdot \sin k\right) \cdot t} \cdot \left(\frac{1}{\tan k} \cdot \frac{\ell}{k}\right)\right)} \]

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	12.0
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-158} \lor \neg \left(k \leq 2.5 \cdot 10^{-44}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 2
Error	4.7
Cost	14025

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{\tan k}\\ \mathbf{if}\;k \leq 2.8 \cdot 10^{-154} \lor \neg \left(k \leq 2.4 \cdot 10^{-56}\right):\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\ell}{k \cdot \left(\sin k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 3
Error	22.3
Cost	8072

\[\begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(-\frac{k}{\ell} \cdot \left(\frac{k}{-\ell} \cdot \left(k \cdot t\right)\right)\right)}\\ \mathbf{elif}\;k \leq 22:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{k \cdot k} + \ell \cdot 0.16666666666666666}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}} + \frac{\ell}{\frac{k}{\frac{\ell}{t}}} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 4
Error	23.3
Cost	7492

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{k \cdot k}}{t}\right)\\ \end{array} \]

Alternative 5
Error	23.4
Cost	7300

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{t}}{{\left(k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 6
Error	23.7
Cost	1224

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -1.28 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-278}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{k \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(k \cdot t\right) \cdot t_1}\\ \end{array} \]

Alternative 7
Error	23.5
Cost	1220

\[\begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-141}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 8
Error	25.2
Cost	960

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

Alternative 9
Error	25.2
Cost	960

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

Alternative 10
Error	25.4
Cost	960

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

Reproduce

herbie shell --seed 2023054 
(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))))