Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 0.3

Time: 27.9s

Precision: binary64

Cost: 13760

Math

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

↓

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

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)) (/ k l)) (/ (/ (/ 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)) / (k / l)) * (((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)) / (k / l)) * (((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)) / (k / l)) * (((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)) / (k / l)) * (((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(Float64(Float64(2.0 / tan(k)) / Float64(k / l)) * 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)) / (k / l)) * (((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[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t), $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 39.8

   \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   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* [=>] 47.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 47.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 39.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 39.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 21.7

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

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] 21.7	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-*r* [=>] 22.4	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
   unpow2 [=>] 22.4	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
   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 8.2

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

6. Simplified 2.8

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

   [Start] 8.2	\[ \frac{2}{\tan k \cdot \frac{-k}{\frac{\ell}{t} \cdot \left(-\frac{\frac{\ell}{k}}{\sin k}\right)}} \]
   associate-/r* [=>] 6.7	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{\frac{-k}{\frac{\ell}{t}}}{-\frac{\frac{\ell}{k}}{\sin k}}}} \]
   associate-/r/ [=>] 1.4	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\frac{-k}{\ell} \cdot t}}{-\frac{\frac{\ell}{k}}{\sin k}}} \]
   associate-/l/ [=>] 2.8	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\color{blue}{\frac{\ell}{\sin k \cdot k}}}} \]
   *-commutative [<=] 2.8	\[ \frac{2}{\tan k \cdot \frac{\frac{-k}{\ell} \cdot t}{-\frac{\ell}{\color{blue}{k \cdot \sin k}}}} \]

7. Applied egg-rr 0.3

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

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-80} \lor \neg \left(k \leq 1.02 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{\sin k}}{t} \cdot \left(\ell \cdot \frac{\frac{2}{k}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.82 \cdot 10^{-144} \lor \neg \left(k \leq 3 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{k}{\ell}} \cdot \frac{\frac{\ell}{k \cdot \sin k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	26.1
Cost	960

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

Alternative 4
Error	25.6
Cost	960

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

Alternative 5
Error	25.5
Cost	960

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

Alternative 6
Error	22.9
Cost	960

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

Alternative 7
Error	22.8
Cost	960

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

Reproduce

herbie shell --seed 2023053 
(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))))