Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 1.3

Time: 29.6s

Precision: binary64

Cost: 13760

Math

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

↓

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

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) (* (/ l k) (/ (/ (/ l k) (sin k)) t)))))

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

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

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

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

Derivation

1. Initial program 47.8

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

2. Simplified 40.1

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

   [Start] 47.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 47.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 47.7	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 47.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 40.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 40.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 22.2

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

4. Simplified 16.4

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

   [Start] 22.2	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r* [=>] 22.9	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
   unpow2 [=>] 22.9	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
   times-frac [=>] 16.4	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
   unpow2 [=>] 16.4	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
   associate-*l* [=>] 16.4	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

5. Applied egg-rr 8.7

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

6. Simplified 3.4

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

   [Start] 8.7	\[ \frac{2}{\tan k \cdot \frac{-k}{\frac{\ell}{t} \cdot \left(-\frac{\frac{\ell}{k}}{\sin k}\right)}} \]
   associate-/r* [=>] 7.2	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{\frac{-k}{\frac{\ell}{t}}}{-\frac{\frac{\ell}{k}}{\sin k}}}} \]
   associate-/r/ [=>] 1.7	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\frac{-k}{\ell} \cdot t}}{-\frac{\frac{\ell}{k}}{\sin k}}} \]
   associate-/l/ [=>] 3.4	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\color{blue}{\frac{\ell}{\sin k \cdot k}}}} \]
   *-commutative [<=] 3.4	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\frac{\ell}{\color{blue}{k \cdot \sin k}}}} \]

7. Applied egg-rr 1.3

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

8. Final simplification 1.3

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

Alternatives

Alternative 1
Error	12.7
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.5 \cdot 10^{-8} \lor \neg \left(k \leq 1.05 \cdot 10^{-63}\right):\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{\frac{2}{\tan k}}{k \cdot k}}{\sin k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	7.0
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -3.5 \cdot 10^{-31} \lor \neg \left(k \leq 4.2 \cdot 10^{-62}\right):\\ \;\;\;\;\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	4.1
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.8 \cdot 10^{-13} \lor \neg \left(k \leq 7.4 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{k \cdot \tan k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	3.8
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -3.1 \cdot 10^{-31} \lor \neg \left(k \leq 4 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{\tan k \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 5
Error	23.2
Cost	1289

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-11} \lor \neg \left(t \leq 10^{+75}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(k \cdot \left(-t\right)\right)}}{t_1} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot t}}{k \cdot t_1}\\ \end{array} \]

Alternative 6
Error	24.2
Cost	1224

\[\begin{array}{l} \mathbf{if}\;k \leq -1.65 \cdot 10^{-93}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot \left(k \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{-89}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot t}\right)\\ \end{array} \]

Alternative 7
Error	24.2
Cost	1224

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-165}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{t_1 \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot \left(k \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Error	24.3
Cost	1224

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+198}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 50:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot t}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot t_1}\\ \end{array} \]

Alternative 9
Error	24.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq -8 \cdot 10^{-154}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k \cdot k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 10
Error	23.2
Cost	1088

\[2 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\right) \]

Alternative 11
Error	22.6
Cost	1088

\[2 \cdot \frac{1}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \left(k \cdot \frac{k}{\ell}\right)} \]

Alternative 12
Error	25.8
Cost	960

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

Alternative 13
Error	25.1
Cost	960

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

Reproduce

herbie shell --seed 2023056 
(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))))