Toniolo and Linder, Equation (10-)

Average Accuracy: 25.9% → 97.5%

Time: 31.8s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

Derivation

1. Initial program 25.9%

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

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] 25.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 25.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 26.0	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 26.0	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   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 65.1%

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

4. Simplified 65.8%

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

   [Start] 65.1	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-/l* [=>] 65.8	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{\sin k \cdot t}}}} \]
   unpow2 [=>] 65.8	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{\sin k \cdot t}}} \]
   unpow2 [=>] 65.8	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot t}}} \]

5. Applied egg-rr 89.1%

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

   [Start] 65.8	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell \cdot \ell}{\sin k \cdot t}}} \]
   times-frac [=>] 73.7	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}} \]
   times-frac [=>] 89.1	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \frac{k}{\frac{\ell}{t}}\right)}} \]

6. Applied egg-rr 97.6%

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

   [Start] 89.1	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \frac{k}{\frac{\ell}{t}}\right)} \]
   associate-/r/ [=>] 97.6	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot t\right)}\right)} \]

7. Applied egg-rr 97.5%

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

   [Start] 97.6	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \left(\frac{k}{\ell} \cdot t\right)\right)} \]
   *-commutative [=>] 97.6	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \color{blue}{\left(t \cdot \frac{k}{\ell}\right)}\right)} \]
   clear-num [=>] 97.5	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \left(t \cdot \color{blue}{\frac{1}{\frac{\ell}{k}}}\right)\right)} \]
   un-div-inv [=>] 97.5	\[ \frac{2}{\tan k \cdot \left(\frac{k}{\frac{\ell}{\sin k}} \cdot \color{blue}{\frac{t}{\frac{\ell}{k}}}\right)} \]

8. Final simplification 97.5%

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

Alternatives

Alternative 1
Accuracy	83.3%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -7.5 \cdot 10^{-30} \lor \neg \left(k \leq 5 \cdot 10^{-27}\right):\\ \;\;\;\;2 \cdot \frac{\ell}{\left(t \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 2
Accuracy	92.7%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.4 \cdot 10^{-42} \lor \neg \left(k \leq 2.3 \cdot 10^{-25}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{\tan k}}{k \cdot \left(\sin k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Accuracy	91.8%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;\ell \leq 1.55 \cdot 10^{-18} \lor \neg \left(\ell \leq 1.35 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\frac{2}{\tan k}}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \tan k} \cdot \frac{\ell}{k \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 4
Accuracy	95.5%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+67} \lor \neg \left(t \leq 2 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{2}{k \cdot \left(\left(k \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\frac{2}{\tan k}}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 5
Accuracy	92.3%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;k \leq -1.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{k \cdot \tan k} \cdot \frac{\ell}{k \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k}}{\tan k}}{k \cdot \left(\sin k \cdot t\right)}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.6%
Cost	13760

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

Alternative 7
Accuracy	59.6%
Cost	960

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

Alternative 8
Accuracy	60.6%
Cost	960

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

Alternative 9
Accuracy	61.4%
Cost	960

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

Alternative 10
Accuracy	63.3%
Cost	960

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

Alternative 11
Accuracy	64.5%
Cost	960

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

Reproduce

herbie shell --seed 2023138 
(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))))