Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.3

Time: 27.3s

Precision: binary64

Cost: 13760

Math

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

↓

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

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
 (/ (/ (* (/ l (tan k)) 2.0) k) (* (* (sin k) (/ k l)) 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 (((l / tan(k)) * 2.0) / k) / ((sin(k) * (k / l)) * 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 = (((l / tan(k)) * 2.0d0) / k) / ((sin(k) * (k / l)) * 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 (((l / Math.tan(k)) * 2.0) / k) / ((Math.sin(k) * (k / l)) * 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 (((l / math.tan(k)) * 2.0) / k) / ((math.sin(k) * (k / l)) * 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(Float64(Float64(Float64(l / tan(k)) * 2.0) / k) / Float64(Float64(sin(k) * Float64(k / l)) * 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 = (((l / tan(k)) * 2.0) / k) / ((sin(k) * (k / l)) * 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[(N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] / k), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.4

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

2. Simplified 39.5

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

   [Start] 47.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 47.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 47.4	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 47.4	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 39.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 39.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 22.1

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

4. Simplified 15.9

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

   [Start] 22.1	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r* [=>] 22.8	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
   unpow2 [=>] 22.8	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
   times-frac [=>] 15.9	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
   unpow2 [=>] 15.9	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
   associate-*l* [=>] 15.9	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

5. Applied egg-rr 7.5

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

6. Simplified 0.3

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

   [Start] 7.5	\[ \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k \cdot \sin k} \cdot \frac{\ell}{k} \]
   *-commutative [=>] 7.5	\[ \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{2}{\tan k} \cdot \frac{\ell}{t}}{k \cdot \sin k}} \]
   associate-/l* [=>] 7.5	\[ \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\frac{k \cdot \sin k}{\frac{\ell}{t}}}} \]
   associate-*r/ [=>] 7.1	\[ \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{k \cdot \sin k}{\frac{\ell}{t}}}} \]
   associate-/r/ [=>] 2.1	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\color{blue}{\frac{k \cdot \sin k}{\ell} \cdot t}} \]
   *-commutative [=>] 2.1	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{\color{blue}{\sin k \cdot k}}{\ell} \cdot t} \]
   *-rgt-identity [<=] 2.1	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\frac{\sin k \cdot \color{blue}{\left(k \cdot 1\right)}}{\ell} \cdot t} \]
   associate-*r/ [<=] 0.3	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\color{blue}{\left(\sin k \cdot \frac{k \cdot 1}{\ell}\right)} \cdot t} \]
   *-rgt-identity [=>] 0.3	\[ \frac{\frac{\ell}{k} \cdot \frac{2}{\tan k}}{\left(\sin k \cdot \frac{\color{blue}{k}}{\ell}\right) \cdot t} \]

7. Applied egg-rr 0.3

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

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	7.2
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -2.5 \cdot 10^{-13} \lor \neg \left(k \leq 5.5 \cdot 10^{-59}\right):\\ \;\;\;\;\ell \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{k}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

Alternative 2
Error	4.0
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -3.1 \cdot 10^{-70} \lor \neg \left(k \leq 3 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\frac{2}{k \cdot \tan k} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{k}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -130000000 \lor \neg \left(k \leq 2 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell \cdot \frac{2}{k}}{\tan k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{2}{k}}{\left(\sin k \cdot \frac{k}{\ell}\right) \cdot t}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13760

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

Alternative 5
Error	22.3
Cost	7360

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

Alternative 6
Error	22.3
Cost	7360

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

Alternative 7
Error	25.2
Cost	960

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

Alternative 8
Error	24.3
Cost	960

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

Alternative 9
Error	22.9
Cost	960

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

Alternative 10
Error	22.4
Cost	960

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

Reproduce

herbie shell --seed 2023018 
(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))))