Toniolo and Linder, Equation (10-)

Average Accuracy: 26.4% → 98.8%

Time: 34.9s

Precision: binary64

Cost: 13760.00

Math

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

↓

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

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

Derivation

1. Initial program 26.4%

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

2. Simplified 38.6%

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

   [Start] 26.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 26.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 26.4	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 26.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 38.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 38.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

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. Applied egg-rr 77.9%

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

5. Simplified 88.1%

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

   [Start] 77.9	\[ \frac{2}{\tan k \cdot \left(\frac{1}{\ell} \cdot \frac{k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)}{\ell}\right)} \]
   associate-/l* [=>] 88.1	\[ \frac{2}{\tan k \cdot \left(\frac{1}{\ell} \cdot \color{blue}{\frac{k}{\frac{\ell}{k \cdot \left(\sin k \cdot t\right)}}}\right)} \]
   *-commutative [=>] 88.1	\[ \frac{2}{\tan k \cdot \left(\frac{1}{\ell} \cdot \frac{k}{\frac{\ell}{k \cdot \color{blue}{\left(t \cdot \sin k\right)}}}\right)} \]

6. Taylor expanded in l around 0 65.5%

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

7. Simplified 97.6%

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

   [Start] 65.5	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   *-commutative [=>] 65.5	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \color{blue}{\left(t \cdot \sin k\right)}}{{\ell}^{2}}} \]
   associate-*r* [=>] 65.5	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}}{{\ell}^{2}}} \]
   unpow2 [=>] 65.5	\[ \frac{2}{\tan k \cdot \frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \sin k}{{\ell}^{2}}} \]
   associate-*l* [=>] 68.9	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \sin k}{{\ell}^{2}}} \]
   associate-*r* [<=] 68.9	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot \left(\left(k \cdot t\right) \cdot \sin k\right)}}{{\ell}^{2}}} \]
   *-commutative [<=] 68.9	\[ \frac{2}{\tan k \cdot \frac{k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot t\right)\right)}}{{\ell}^{2}}} \]
   *-commutative [=>] 68.9	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left(\sin k \cdot \left(k \cdot t\right)\right) \cdot k}}{{\ell}^{2}}} \]
   *-commutative [<=] 68.9	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot \left(\sin k \cdot \left(k \cdot t\right)\right)}}{{\ell}^{2}}} \]
   unpow2 [=>] 68.9	\[ \frac{2}{\tan k \cdot \frac{k \cdot \left(\sin k \cdot \left(k \cdot t\right)\right)}{\color{blue}{\ell \cdot \ell}}} \]
   times-frac [=>] 89.9	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{\sin k \cdot \left(k \cdot t\right)}{\ell}\right)}} \]
   *-commutative [=>] 89.9	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{\sin k \cdot \left(k \cdot t\right)}{\ell} \cdot \frac{k}{\ell}\right)}} \]
   associate-/l* [=>] 91.2	\[ \frac{2}{\tan k \cdot \left(\color{blue}{\frac{\sin k}{\frac{\ell}{k \cdot t}}} \cdot \frac{k}{\ell}\right)} \]
   associate-/r* [=>] 97.6	\[ \frac{2}{\tan k \cdot \left(\frac{\sin k}{\color{blue}{\frac{\frac{\ell}{k}}{t}}} \cdot \frac{k}{\ell}\right)} \]

8. Applied egg-rr 35.7%

   \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\frac{k}{\ell} \cdot \tan k\right)\right)} - 1}} \]

9. Simplified 98.8%

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

   [Start] 35.7	\[ \frac{2}{e^{\mathsf{log1p}\left(\left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\frac{k}{\ell} \cdot \tan k\right)\right)} - 1} \]
   expm1-def [=>] 59.5	\[ \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\frac{k}{\ell} \cdot \tan k\right)\right)\right)}} \]
   expm1-log1p [=>] 98.8	\[ \frac{2}{\color{blue}{\left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right) \cdot \left(\frac{k}{\ell} \cdot \tan k\right)}} \]

10. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	93.0%
Cost	14290.00

\[\begin{array}{l} \mathbf{if}\;k \leq -4.2 \cdot 10^{-11} \lor \neg \left(k \leq -7.5 \cdot 10^{-155}\right) \land \left(k \leq 7.5 \cdot 10^{-155} \lor \neg \left(k \leq 6.7 \cdot 10^{-18}\right)\right):\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\frac{\ell}{k}}{\frac{\sin k \cdot \left(t \cdot k\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \]

Alternative 2
Accuracy	92.5%
Cost	14288.00

\[\begin{array}{l} t_1 := \frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}}\\ t_2 := \frac{2}{\tan k}\\ t_3 := t_2 \cdot \frac{\frac{\ell}{k}}{\frac{\sin k \cdot \left(t \cdot k\right)}{\ell}}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.18 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{k} \cdot \frac{\ell}{k \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 3
Accuracy	81.7%
Cost	14025.00

\[\begin{array}{l} \mathbf{if}\;k \leq -3.5 \cdot 10^{-10} \lor \neg \left(k \leq 4.9 \cdot 10^{-18}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \]

Alternative 4
Accuracy	88.9%
Cost	14025.00

\[\begin{array}{l} \mathbf{if}\;k \leq -700 \lor \neg \left(k \leq 2.35 \cdot 10^{-17}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\frac{2}{k}}{\tan k} \cdot \frac{\ell}{k \cdot \left(t \cdot \sin k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \]

Alternative 5
Accuracy	89.1%
Cost	14025.00

\[\begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-10} \lor \neg \left(k \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{2 \cdot \ell}{k}}{\tan k \cdot \left(\sin k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}}\\ \end{array} \]

Alternative 6
Accuracy	94.6%
Cost	14020.00

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+255}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\frac{\ell}{t \cdot k}}} \cdot \frac{2}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	92.8%
Cost	13892.00

\[\begin{array}{l} t_1 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{\sin k}{\frac{\ell}{t \cdot k}}} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{k} \cdot \frac{\ell}{k \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 8
Accuracy	97.6%
Cost	13760.00

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

Alternative 9
Accuracy	61.3%
Cost	7624.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\frac{2}{t}}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot \frac{t}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{k \cdot k}\right)\\ \end{array} \]

Alternative 10
Accuracy	61.5%
Cost	7624.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{k \cdot \left(t \cdot k\right)}{\ell}}}{k}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-82}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k \cdot \left(t \cdot \sin k\right)} \cdot \frac{2}{k \cdot k}\right)\\ \end{array} \]

Alternative 11
Accuracy	61.2%
Cost	7560.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+53}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\frac{2}{t}}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot \frac{t}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2 \cdot {k}^{-2}}{\frac{k \cdot \left(t \cdot k\right)}{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	61.9%
Cost	7488.00

\[\frac{2}{\frac{t \cdot \frac{k \cdot k}{\ell}}{\ell \cdot \frac{\cos k}{k \cdot k}}} \]

Alternative 13
Accuracy	61.2%
Cost	1352.00

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+46}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\frac{2}{t}}}}{k \cdot k}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot \frac{t}{\ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{1}{k \cdot k}}{\frac{t \cdot 0.5}{\frac{\ell}{k \cdot k}}}\\ \end{array} \]

Alternative 14
Accuracy	61.3%
Cost	1225.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+54} \lor \neg \left(t \leq 4.3 \cdot 10^{-78}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{\frac{k \cdot k}{\frac{2}{t}}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot \frac{t}{\ell}}}{k \cdot k}\\ \end{array} \]

Alternative 15
Accuracy	59.3%
Cost	960.00

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

Alternative 16
Accuracy	59.3%
Cost	960.00

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

Reproduce

herbie shell --seed 2023096 
(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))))