Toniolo and Linder, Equation (10-)

Average Error: 47.2 → 12.2

Time: 38.4s

Precision: binary64

Cost: 20096

Math

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

↓

\[\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{\frac{\ell}{\sin k \cdot 0.5}}} \]

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
 (/ (/ (/ l (pow k 2.0)) t) (/ (tan k) (/ l (* (sin k) 0.5)))))

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 ((l / pow(k, 2.0)) / t) / (tan(k) / (l / (sin(k) * 0.5)));
}

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 = ((l / (k ** 2.0d0)) / t) / (tan(k) / (l / (sin(k) * 0.5d0)))
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 ((l / Math.pow(k, 2.0)) / t) / (Math.tan(k) / (l / (Math.sin(k) * 0.5)));
}

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 ((l / math.pow(k, 2.0)) / t) / (math.tan(k) / (l / (math.sin(k) * 0.5)))

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(Float64(Float64(l / (k ^ 2.0)) / t) / Float64(tan(k) / Float64(l / Float64(sin(k) * 0.5))))
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 = ((l / (k ^ 2.0)) / t) / (tan(k) / (l / (sin(k) * 0.5)));
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[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.2

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

2. Simplified 39.4

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

   [Start] 47.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   rational.json-simplify-46 [=>] 47.2	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
   rational.json-simplify-48 [=>] 39.4	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
   metadata-eval [=>] 39.4	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
   rational.json-simplify-4 [=>] 39.4	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

3. Applied egg-rr 37.6

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

4. Simplified 36.0

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

   [Start] 37.6	\[ \frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} + 0 \]
   rational.json-simplify-4 [=>] 37.6	\[ \color{blue}{\frac{2}{\left(\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
   rational.json-simplify-46 [=>] 37.6	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}{{\left(\frac{k}{t}\right)}^{2}}} \]
   rational.json-simplify-46 [=>] 37.6	\[ \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\tan k}{\ell \cdot \frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   rational.json-simplify-46 [=>] 37.2	\[ \frac{\frac{\frac{2}{\sin k}}{\color{blue}{\frac{\frac{\tan k}{\ell}}{\frac{\ell}{{t}^{3}}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   rational.json-simplify-61 [=>] 36.9	\[ \frac{\color{blue}{\frac{\frac{\ell}{{t}^{3}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}}}{{\left(\frac{k}{t}\right)}^{2}} \]
   rational.json-simplify-44 [=>] 36.0	\[ \color{blue}{\frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\frac{\frac{\tan k}{\ell}}{\frac{2}{\sin k}}}} \]
   rational.json-simplify-47 [=>] 36.0	\[ \frac{\frac{\frac{\ell}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}}}{\color{blue}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}}} \]

5. Taylor expanded in l around 0 15.5

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

6. Applied egg-rr 15.5

   \[\leadsto \frac{\color{blue}{\frac{\ell}{{k}^{2} \cdot t} + 0}}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}} \]

7. Simplified 12.2

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

   [Start] 15.5	\[ \frac{\frac{\ell}{{k}^{2} \cdot t} + 0}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}} \]
   rational.json-simplify-4 [=>] 15.5	\[ \frac{\color{blue}{\frac{\ell}{{k}^{2} \cdot t}}}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}} \]
   rational.json-simplify-46 [=>] 12.2	\[ \frac{\color{blue}{\frac{\frac{\ell}{{k}^{2}}}{t}}}{\frac{\tan k}{\ell \cdot \frac{2}{\sin k}}} \]

8. Applied egg-rr 12.2

   \[\leadsto \frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{\color{blue}{\frac{\ell}{\sin k \cdot 0.5}}}} \]

9. Final simplification 12.2

   \[\leadsto \frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{\frac{\ell}{\sin k \cdot 0.5}}} \]

Alternatives

Alternative 1
Error	14.1
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{2 \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.1
Cost	20360

\[\begin{array}{l} \mathbf{if}\;k \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{\sin k} \cdot \frac{\ell}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 1.52 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{2 \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left({k}^{2} \cdot t\right)}\right)\\ \end{array} \]

Alternative 3
Error	12.4
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \frac{\frac{\frac{\ell}{t}}{\sin k \cdot \left(\tan k \cdot 0.5\right)}}{{k}^{2}}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{2 \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	12.4
Cost	20360

\[\begin{array}{l} \mathbf{if}\;k \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \frac{\frac{\frac{\ell}{t}}{\sin k \cdot \left(\tan k \cdot 0.5\right)}}{{k}^{2}}\\ \mathbf{elif}\;k \leq 7.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2}}}{t}}{\frac{\tan k}{2 \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{\sin k} \cdot \frac{\frac{\ell}{t}}{{k}^{2} \cdot \tan k}\right) \cdot \ell\\ \end{array} \]

Alternative 5
Error	16.2
Cost	20096

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

Alternative 6
Error	16.1
Cost	20096

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

Alternative 7
Error	12.9
Cost	20096

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

Alternative 8
Error	12.2
Cost	20096

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

Alternative 9
Error	26.1
Cost	13696

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

Alternative 10
Error	24.0
Cost	13696

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

Alternative 11
Error	24.0
Cost	13696

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

Alternative 12
Error	25.5
Cost	13632

\[\frac{\frac{\ell}{{k}^{2} \cdot t}}{0.5 \cdot \frac{{k}^{2}}{\ell}} \]

Alternative 13
Error	28.9
Cost	7424

\[\frac{1}{2 \cdot \left(0.5 \cdot \left({k}^{4} \cdot \frac{t}{\ell}\right)\right)} \cdot \left(\ell + \ell\right) \]

Alternative 14
Error	29.5
Cost	7040

\[\ell \cdot \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \]

Reproduce

herbie shell --seed 2023068 
(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))))