Toniolo and Linder, Equation (10-)

Average Accuracy: 25.8% → 98.5%

Time: 29.6s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

Derivation

1. Initial program 25.8%

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

2. Simplified 38.2%

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

   [Start] 25.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 25.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 25.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 25.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 38.2	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 38.2	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 65.5%

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

4. Simplified 74.6%

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

   [Start] 65.5	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r* [=>] 64.3	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
   unpow2 [=>] 64.3	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
   times-frac [=>] 74.6	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
   unpow2 [=>] 74.6	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
   associate-*l* [=>] 74.6	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

5. Applied egg-rr 90.6%

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

   [Start] 74.6	\[ \frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)} \]
   associate-/r* [=>] 74.6	\[ \color{blue}{\frac{\frac{2}{\tan k}}{\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}}} \]
   *-commutative [=>] 74.6	\[ \frac{\frac{2}{\tan k}}{\color{blue}{\frac{t}{\ell} \cdot \frac{k \cdot \left(k \cdot \sin k\right)}{\ell}}} \]
   associate-/l* [=>] 83.1	\[ \frac{\frac{2}{\tan k}}{\frac{t}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k \cdot \sin k}}}} \]
   associate-*r/ [=>] 89.0	\[ \frac{\frac{2}{\tan k}}{\color{blue}{\frac{\frac{t}{\ell} \cdot k}{\frac{\ell}{k \cdot \sin k}}}} \]
   associate-/r/ [=>] 89.8	\[ \color{blue}{\frac{\frac{2}{\tan k}}{\frac{t}{\ell} \cdot k} \cdot \frac{\ell}{k \cdot \sin k}} \]
   associate-/r* [=>] 90.6	\[ \frac{\frac{2}{\tan k}}{\frac{t}{\ell} \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sin k}} \]

6. Taylor expanded in t around 0 93.0%

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

7. Simplified 98.5%

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

   [Start] 93.0	\[ \frac{\frac{2}{\tan k}}{\frac{k \cdot t}{\ell}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
   *-commutative [<=] 93.0	\[ \frac{\frac{2}{\tan k}}{\frac{\color{blue}{t \cdot k}}{\ell}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
   associate-*r/ [<=] 98.5	\[ \frac{\frac{2}{\tan k}}{\color{blue}{t \cdot \frac{k}{\ell}}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]

8. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	92.8%
Cost	14020

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

Alternative 2
Accuracy	93.6%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\frac{2}{\tan k}}{k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k \cdot t}}{\tan k}\right)\\ \end{array} \]

Alternative 3
Accuracy	93.2%
Cost	13760

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

Alternative 4
Accuracy	65.3%
Cost	7360

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

Alternative 5
Accuracy	59.9%
Cost	1225

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;\ell \leq -7.5 \cdot 10^{-227} \lor \neg \left(\ell \leq 5 \cdot 10^{-254}\right):\\ \;\;\;\;\ell \cdot \left(t_1 \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \end{array} \]

Alternative 6
Accuracy	59.9%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1300000000 \lor \neg \left(t \leq -2.1 \cdot 10^{-178}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot k}\right)\\ \end{array} \]

Alternative 7
Accuracy	60.5%
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := t \cdot \left(k \cdot k\right)\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;\ell \cdot \left(t_1 \cdot \frac{2}{t_2}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-228}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{k \cdot k} \cdot \frac{\ell}{t_2}\\ \end{array} \]

Alternative 8
Accuracy	59.4%
Cost	960

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

Alternative 9
Accuracy	63.7%
Cost	960

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

Alternative 10
Accuracy	65.0%
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 2023135 
(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))))