Toniolo and Linder, Equation (10-)

Average Error: 73.89% → 1.84%

Time: 31.2s

Precision: binary64

Cost: 13760

Math

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

↓

\[\frac{2}{\frac{\tan k}{\frac{\ell}{k} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{t}}} \]

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 (/ (tan k) (* (/ l k) (/ (/ (/ l k) (sin k)) t)))))

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 / (tan(k) / ((l / k) * (((l / k) / sin(k)) / t)));
}

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 / (tan(k) / ((l / k) * (((l / k) / sin(k)) / t)))
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 / (Math.tan(k) / ((l / k) * (((l / k) / Math.sin(k)) / t)));
}

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 / (math.tan(k) / ((l / k) * (((l / k) / math.sin(k)) / t)))

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(tan(k) / Float64(Float64(l / k) * Float64(Float64(Float64(l / k) / sin(k)) / t))))
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 / (tan(k) / ((l / k) * (((l / k) / sin(k)) / t)));
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[Tan[k], $MachinePrecision] / N[(N[(l / k), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.89

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

2. Simplified 61.5

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

   [Start] 73.89	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 73.89	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 73.87	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 73.87	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 61.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 61.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 34.42

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

4. Simplified 24.94

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

   [Start] 34.42	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r* [=>] 35.63	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
   unpow2 [=>] 35.63	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
   times-frac [=>] 24.94	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
   unpow2 [=>] 24.94	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
   associate-*l* [=>] 24.94	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

5. Applied egg-rr 12.69

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

6. Simplified 4.84

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

   [Start] 12.69	\[ \frac{2}{\tan k \cdot \frac{-k}{\frac{\ell}{t} \cdot \left(-\frac{\frac{\ell}{k}}{\sin k}\right)}} \]
   associate-/r* [=>] 10.38	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{\frac{-k}{\frac{\ell}{t}}}{-\frac{\frac{\ell}{k}}{\sin k}}}} \]
   associate-/r/ [=>] 2.58	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\frac{-k}{\ell} \cdot t}}{-\frac{\frac{\ell}{k}}{\sin k}}} \]
   associate-/l/ [=>] 4.84	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\color{blue}{\frac{\ell}{\sin k \cdot k}}}} \]
   *-commutative [<=] 4.84	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\frac{\ell}{\color{blue}{k \cdot \sin k}}}} \]

7. Applied egg-rr 1.84

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

8. Final simplification 1.84

   \[\leadsto \frac{2}{\frac{\tan k}{\frac{\ell}{k} \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{t}}} \]

Alternatives

Alternative 1
Error	6.52%
Cost	14280

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-301}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{k \cdot t} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\frac{2}{k \cdot \tan k}}{\frac{t}{\ell}}\\ \end{array} \]

Alternative 2
Error	7.07%
Cost	14156

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ t_2 := \frac{2}{\tan k}\\ t_3 := \frac{t_2}{k \cdot \frac{t}{\ell}} \cdot \frac{\ell}{k \cdot \sin k}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-144}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq 1.18 \cdot 10^{+196}:\\ \;\;\;\;\frac{t_2}{k \cdot t} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	7.06%
Cost	14156

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{\sin k}\\ t_2 := \frac{2}{\tan k}\\ t_3 := \frac{t_2}{k \cdot \frac{t}{\ell}}\\ t_4 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-143}:\\ \;\;\;\;t_3 \cdot \frac{\ell}{k \cdot \sin k}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{2}{t_4 \cdot \left(t \cdot t_4\right)}\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+190}:\\ \;\;\;\;\frac{t_2}{k \cdot t} \cdot \left(\ell \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_3\\ \end{array} \]

Alternative 4
Error	6.36%
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.2 \cdot 10^{-122} \lor \neg \left(k \leq 3.8 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\frac{2}{\tan k}}{k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	5.77%
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2.05 \cdot 10^{-116} \lor \neg \left(k \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\frac{2}{k \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Error	6.35%
Cost	14024

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\frac{2}{\tan k}}{k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\frac{2}{k \cdot \tan k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7
Error	1.76%
Cost	13760

\[\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{2}{\tan k}}{\frac{t}{\frac{\ell}{k}}} \]

Alternative 8
Error	40.52%
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 9
Error	37.85%
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_1 \cdot \frac{t_1}{t \cdot 0.5} \end{array} \]

Alternative 10
Error	35.26%
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 2023121 
(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))))