Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 95.9%

Time: 18.1s

Precision: binary64

Cost: 13760

• 40.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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

Derivation

1. Initial program 28.7%

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

2. Simplified 43.3%

   \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
   Step-by-step derivation

   [Start] 28.7%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 28.7%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 28.7%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   associate-/r* [=>] 28.7%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
   associate-/r/ [=>] 28.6%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 28.6%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   times-frac [=>] 29.5%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
   +-commutative [=>] 29.5%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   associate--l+ [=>] 39.5%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   metadata-eval [=>] 39.5%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   +-rgt-identity [=>] 39.5%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   times-frac [=>] 43.3%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Taylor expanded in t around 0 81.8%

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

4. Simplified 81.8%

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

   [Start] 81.8%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   unpow2 [=>] 81.8%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

5. Applied egg-rr 84.1%

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

   [Start] 81.8%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   associate-*l/ [=>] 81.5%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
   associate-*l* [=>] 84.1%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

6. Simplified 90.4%

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

   [Start] 84.1%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
   associate-*l/ [<=] 84.4%	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
   associate-*r* [=>] 90.4%	\[ \color{blue}{\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]
   *-commutative [=>] 90.4%	\[ \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right)} \]
   *-commutative [=>] 90.4%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)} \]

7. Taylor expanded in l around 0 85.9%

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

8. Simplified 93.9%

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

   [Start] 85.9%	\[ \frac{\ell}{\tan k} \cdot \left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}\right) \]
   associate-*r/ [=>] 85.9%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}} \]
   *-commutative [=>] 85.9%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot t\right) \cdot {k}^{2}}} \]
   associate-*r* [<=] 85.8%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}} \]
   *-commutative [<=] 85.8%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \]
   unpow2 [=>] 85.8%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]
   associate-*r* [<=] 88.5%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}} \]
   *-commutative [<=] 88.5%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \sin k}} \]
   associate-*l* [=>] 88.5%	\[ \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{k \cdot \left(\left(k \cdot t\right) \cdot \sin k\right)}} \]
   times-frac [=>] 93.9%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\left(\frac{2}{k} \cdot \frac{\ell}{\left(k \cdot t\right) \cdot \sin k}\right)} \]
   *-commutative [=>] 93.9%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{\color{blue}{\sin k \cdot \left(k \cdot t\right)}}\right) \]

9. Taylor expanded in l around 0 93.8%

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

10. Simplified 95.8%

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

    [Start] 93.8%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot t\right)}\right) \]
    associate-/r* [=>] 95.8%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{2}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sin k \cdot t}}\right) \]

11. Final simplification 95.8%

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

Alternatives

Alternative 1
Accuracy	95.9%
Cost	13760

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

Alternative 2
Accuracy	82.7%
Cost	14020

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;t_1 \cdot \left(\frac{2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(t_1 \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.6%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{2}{k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \ell}{\tan k \cdot \sin k}\\ \end{array} \]

Alternative 4
Accuracy	87.2%
Cost	13760

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

Alternative 5
Accuracy	93.9%
Cost	13760

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

Alternative 6
Accuracy	75.6%
Cost	8009

\[\begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-260} \lor \neg \left(t \leq 4.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{k \cdot k}}{\sin k \cdot t}\right) \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(\ell \cdot k\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	75.1%
Cost	7360

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

Alternative 8
Accuracy	73.7%
Cost	960

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

Alternative 9
Accuracy	73.7%
Cost	960

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

Reproduce

herbie shell --seed 2023272 
(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))))