Toniolo and Linder, Equation (10-)

Average Error: 47.0 → 0.6

Time: 40.2s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

Derivation

1. Initial program 47.0

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

2. Simplified 39.4

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

   [Start] 47.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 47.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 46.9	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 46.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 39.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 39.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 21.9

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

4. Simplified 20.0

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

   [Start] 21.9	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-/l* [=>] 22.2	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{\sin k \cdot t}}}} \]
   unpow2 [=>] 22.2	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{\sin k \cdot t}}} \]
   associate-/l* [=>] 20.0	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{k}{\frac{\frac{{\ell}^{2}}{\sin k \cdot t}}{k}}}} \]
   unpow2 [=>] 20.0	\[ \frac{2}{\tan k \cdot \frac{k}{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot t}}{k}}} \]

5. Applied egg-rr 10.6

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

6. Simplified 0.6

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

   [Start] 10.6	\[ \frac{2}{\frac{\tan k \cdot \left(-k\right)}{\left(-\frac{\ell}{\sin k}\right) \cdot \frac{\ell}{k \cdot t}}} \]
   *-commutative [=>] 10.6	\[ \frac{2}{\frac{\tan k \cdot \left(-k\right)}{\color{blue}{\frac{\ell}{k \cdot t} \cdot \left(-\frac{\ell}{\sin k}\right)}}} \]
   times-frac [=>] 4.6	\[ \frac{2}{\color{blue}{\frac{\tan k}{\frac{\ell}{k \cdot t}} \cdot \frac{-k}{-\frac{\ell}{\sin k}}}} \]
   associate-/r* [=>] 1.1	\[ \frac{2}{\frac{\tan k}{\color{blue}{\frac{\frac{\ell}{k}}{t}}} \cdot \frac{-k}{-\frac{\ell}{\sin k}}} \]
   associate-/r/ [=>] 0.6	\[ \frac{2}{\color{blue}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right)} \cdot \frac{-k}{-\frac{\ell}{\sin k}}} \]
   neg-mul-1 [=>] 0.6	\[ \frac{2}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right) \cdot \frac{\color{blue}{-1 \cdot k}}{-\frac{\ell}{\sin k}}} \]
   neg-mul-1 [=>] 0.6	\[ \frac{2}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right) \cdot \frac{-1 \cdot k}{\color{blue}{-1 \cdot \frac{\ell}{\sin k}}}} \]
   times-frac [=>] 0.6	\[ \frac{2}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right) \cdot \color{blue}{\left(\frac{-1}{-1} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)}} \]
   metadata-eval [=>] 0.6	\[ \frac{2}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right) \cdot \left(\color{blue}{1} \cdot \frac{k}{\frac{\ell}{\sin k}}\right)} \]
   associate-/r/ [=>] 0.6	\[ \frac{2}{\left(\frac{\tan k}{\frac{\ell}{k}} \cdot t\right) \cdot \left(1 \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \sin k\right)}\right)} \]

7. Final simplification 0.6

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

Alternatives

Alternative 1
Error	6.3
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-306}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\ell}{\sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)}}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	14288

\[\begin{array}{l} t_1 := \frac{2}{\tan k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ t_2 := \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-187}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	9.3
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sin k\\ t_2 := \frac{2}{\tan k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot t_1\right)\right)}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{t_1}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-182}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;\ell \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	6.3
Cost	14280

\[\begin{array}{l} t_1 := t \cdot \sin k\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-306}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 4 \cdot 10^{+307}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t_1}}}\\ \end{array} \]

Alternative 5
Error	10.3
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-12} \lor \neg \left(k \leq 2.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(\frac{\sin k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \end{array} \]

Alternative 6
Error	3.2
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+114} \lor \neg \left(t \leq 1.7 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k}{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot \sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{\frac{k}{\ell} \cdot \sin k}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	1988

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-306}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\frac{1}{t}}{k \cdot k} + \frac{\frac{\ell}{t}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\right)\\ \end{array} \]

Alternative 8
Error	23.3
Cost	1088

\[2 \cdot \frac{1}{\frac{k \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{k}}{k \cdot t}}} \]

Alternative 9
Error	25.7
Cost	960

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

Alternative 10
Error	25.0
Cost	960

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

Alternative 11
Error	24.6
Cost	960

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

Alternative 12
Error	23.2
Cost	960

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

Alternative 13
Error	23.5
Cost	960

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

Alternative 14
Error	33.6
Cost	704

\[\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

Reproduce

herbie shell --seed 2023060 
(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))))