Toniolo and Linder, Equation (10-)

Average Accuracy: 24.7% → 98.9%

Time: 31.8s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

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

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

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

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

Derivation

1. Initial program 24.7%

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

2. Simplified 37.3%

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

   [Start] 24.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 24.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 24.7	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 24.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 37.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 37.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 65.7%

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

4. Simplified 72.5%

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

   [Start] 65.7	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-/l* [=>] 65.6	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{\sin k \cdot t}}}} \]
   unpow2 [=>] 65.6	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{\sin k \cdot t}}} \]
   unpow2 [=>] 65.6	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot t}}} \]
   associate-/l* [=>] 72.5	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{\frac{\sin k \cdot t}{\ell}}}}} \]

5. Applied egg-rr 70.5%

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

   [Start] 72.5	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{\sin k \cdot t}{\ell}}}} \]
   frac-2neg [=>] 72.5	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{-k \cdot k}{-\frac{\ell}{\frac{\sin k \cdot t}{\ell}}}}} \]
   associate-*r/ [=>] 70.5	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot \left(-k \cdot k\right)}{-\frac{\ell}{\frac{\sin k \cdot t}{\ell}}}}} \]
   distribute-rgt-neg-in [=>] 70.5	\[ \frac{2}{\frac{\tan k \cdot \color{blue}{\left(k \cdot \left(-k\right)\right)}}{-\frac{\ell}{\frac{\sin k \cdot t}{\ell}}}} \]
   associate-/r/ [=>] 70.5	\[ \frac{2}{\frac{\tan k \cdot \left(k \cdot \left(-k\right)\right)}{-\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \ell}}} \]
   distribute-rgt-neg-in [=>] 70.5	\[ \frac{2}{\frac{\tan k \cdot \left(k \cdot \left(-k\right)\right)}{\color{blue}{\frac{\ell}{\sin k \cdot t} \cdot \left(-\ell\right)}}} \]

6. Simplified 96.1%

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

   [Start] 70.5	\[ \frac{2}{\frac{\tan k \cdot \left(k \cdot \left(-k\right)\right)}{\frac{\ell}{\sin k \cdot t} \cdot \left(-\ell\right)}} \]
   *-commutative [<=] 70.5	\[ \frac{2}{\frac{\color{blue}{\left(k \cdot \left(-k\right)\right) \cdot \tan k}}{\frac{\ell}{\sin k \cdot t} \cdot \left(-\ell\right)}} \]
   associate-*l* [=>] 70.5	\[ \frac{2}{\frac{\color{blue}{k \cdot \left(\left(-k\right) \cdot \tan k\right)}}{\frac{\ell}{\sin k \cdot t} \cdot \left(-\ell\right)}} \]
   times-frac [=>] 89.7	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{\sin k \cdot t}} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}}} \]
   associate-/r/ [=>] 95.3	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)\right)} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   associate-*l/ [=>] 88.9	\[ \frac{2}{\color{blue}{\frac{k \cdot \left(\sin k \cdot t\right)}{\ell}} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   associate-*r/ [<=] 90.0	\[ \frac{2}{\color{blue}{\left(k \cdot \frac{\sin k \cdot t}{\ell}\right)} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   associate-*l/ [<=] 90.7	\[ \frac{2}{\left(k \cdot \color{blue}{\left(\frac{\sin k}{\ell} \cdot t\right)}\right) \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   *-commutative [=>] 90.7	\[ \frac{2}{\color{blue}{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot k\right)} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   *-commutative [=>] 90.7	\[ \frac{2}{\left(\color{blue}{\left(t \cdot \frac{\sin k}{\ell}\right)} \cdot k\right) \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   associate-*l* [=>] 96.1	\[ \frac{2}{\color{blue}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right)} \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]

7. Applied egg-rr 98.9%

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

   [Start] 96.1	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\left(-k\right) \cdot \tan k}{-\ell}} \]
   *-commutative [=>] 96.1	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\color{blue}{\tan k \cdot \left(-k\right)}}{-\ell}} \]
   associate-/l* [=>] 99.0	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \color{blue}{\frac{\tan k}{\frac{-\ell}{-k}}}} \]
   add-sqr-sqrt [=>] 49.1	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}{-k}}} \]
   sqrt-unprod [=>] 60.2	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}{-k}}} \]
   sqr-neg [=>] 60.2	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\sqrt{\color{blue}{\ell \cdot \ell}}}{-k}}} \]
   sqrt-unprod [<=] 23.8	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{-k}}} \]
   add-sqr-sqrt [<=] 47.3	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\color{blue}{\ell}}{-k}}} \]
   add-sqr-sqrt [=>] 23.3	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\ell}{\color{blue}{\sqrt{-k} \cdot \sqrt{-k}}}}} \]
   sqrt-unprod [=>] 66.7	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\ell}{\color{blue}{\sqrt{\left(-k\right) \cdot \left(-k\right)}}}}} \]
   sqr-neg [=>] 66.7	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\ell}{\sqrt{\color{blue}{k \cdot k}}}}} \]
   sqrt-unprod [<=] 52.2	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\ell}{\color{blue}{\sqrt{k} \cdot \sqrt{k}}}}} \]
   add-sqr-sqrt [<=] 99.0	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \frac{\tan k}{\frac{\ell}{\color{blue}{k}}}} \]
   associate-/r/ [=>] 98.9	\[ \frac{2}{\left(t \cdot \left(\frac{\sin k}{\ell} \cdot k\right)\right) \cdot \color{blue}{\left(\frac{\tan k}{\ell} \cdot k\right)}} \]

8. Final simplification 98.9%

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

Alternatives

Alternative 1
Accuracy	88.8%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.6 \cdot 10^{-110} \lor \neg \left(k \leq 1.8 \cdot 10^{-30}\right):\\ \;\;\;\;2 \cdot \frac{\ell}{\left(k \cdot \tan k\right) \cdot \left(k \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 2
Accuracy	92.4%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-8} \lor \neg \left(k \leq 1.46 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{2}{k \cdot \tan k} \cdot \left(\frac{\ell}{t \cdot \sin k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Accuracy	94.5%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-97} \lor \neg \left(k \leq 8.2 \cdot 10^{-31}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot \tan k}}{k \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 4
Accuracy	94.7%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-124} \lor \neg \left(t \leq 6.5 \cdot 10^{-208}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{t \cdot \sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\sin k}}{t \cdot k}}{\tan k \cdot 0.5}\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	13760

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

Alternative 6
Accuracy	59.5%
Cost	960

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

Alternative 7
Accuracy	59.6%
Cost	960

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

Alternative 8
Accuracy	60.2%
Cost	960

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

Alternative 9
Accuracy	61.5%
Cost	960

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

Alternative 10
Accuracy	63.8%
Cost	960

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

Alternative 11
Accuracy	63.7%
Cost	960

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

Alternative 12
Accuracy	64.6%
Cost	960

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

Reproduce

herbie shell --seed 2023141 
(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))))