Toniolo and Linder, Equation (10-)

Average Accuracy: 25.7% → 98.7%

Time: 32.4s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

Derivation

1. Initial program 25.7%

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

2. Simplified 37.9%

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

   [Start] 25.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 25.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 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+ [=>] 37.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 37.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 64.6%

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

4. Applied egg-rr 67.6%

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

   [Start] 64.6	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r/ [=>] 63.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}{{\ell}^{2}}}} \]
   clear-num [=>] 63.9	\[ \frac{2}{\color{blue}{\frac{1}{\frac{{\ell}^{2}}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}}} \]
   associate-/r/ [=>] 64.0	\[ \color{blue}{\frac{2}{1} \cdot \frac{{\ell}^{2}}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   metadata-eval [=>] 64.0	\[ \color{blue}{2} \cdot \frac{{\ell}^{2}}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
   unpow2 [=>] 64.0	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
   *-commutative [=>] 64.0	\[ 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}} \]
   unpow2 [=>] 64.0	\[ 2 \cdot \frac{\ell \cdot \ell}{\left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k} \]
   associate-*l* [=>] 67.6	\[ 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right)} \cdot \tan k} \]

5. Simplified 79.2%

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

   [Start] 67.6	\[ 2 \cdot \frac{\ell \cdot \ell}{\left(k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)\right) \cdot \tan k} \]
   times-frac [=>] 79.2	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)} \cdot \frac{\ell}{\tan k}\right)} \]
   *-commutative [=>] 79.2	\[ 2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \cdot \frac{\ell}{\tan k}\right) \]

6. Applied egg-rr 98.7%

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

   [Start] 79.2	\[ 2 \cdot \left(\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)} \cdot \frac{\ell}{\tan k}\right) \]
   associate-*r/ [=>] 78.3	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)} \cdot \ell}{\tan k}} \]
   associate-/l* [=>] 79.2	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\frac{\tan k}{\ell}}} \]
   associate-/r* [=>] 90.1	\[ 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k}}{k \cdot \left(t \cdot \sin k\right)}}}{\frac{\tan k}{\ell}} \]
   associate-/l/ [<=] 93.3	\[ 2 \cdot \frac{\color{blue}{\frac{\frac{\frac{\ell}{k}}{t \cdot \sin k}}{k}}}{\frac{\tan k}{\ell}} \]
   associate-/l/ [=>] 98.7	\[ 2 \cdot \color{blue}{\frac{\frac{\frac{\ell}{k}}{t \cdot \sin k}}{\frac{\tan k}{\ell} \cdot k}} \]

7. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	88.7%
Cost	14156

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := k \cdot \left(t \cdot \sin k\right)\\ t_3 := 2 \cdot \frac{\frac{\ell}{k} \cdot t_1}{t_2}\\ \mathbf{if}\;k \leq -0.00045:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k \cdot t}}{k \cdot \frac{\tan k}{\ell}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+95}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\ell}{k \cdot t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Accuracy	84.4%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00045 \lor \neg \left(k \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k \cdot t}}{k \cdot \frac{\tan k}{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	62.8%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-35} \lor \neg \left(t \leq 1.35 \cdot 10^{-44}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{k \cdot \frac{\tan k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}}{k \cdot k}\\ \end{array} \]

Alternative 4
Accuracy	63.1%
Cost	7360

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

Alternative 5
Accuracy	63.0%
Cost	7360

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

Alternative 6
Accuracy	59.9%
Cost	1220

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

Alternative 7
Accuracy	60.1%
Cost	1220

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-244}:\\ \;\;\;\;\frac{2}{t} \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}}{k}\\ \end{array} \]

Alternative 8
Accuracy	58.0%
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \frac{2}{t} \cdot \left(t_1 \cdot t_1\right) \end{array} \]

Reproduce

herbie shell --seed 2023137 
(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))))