Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.3% → 99.0%

Time: 25.9s

Precision: binary64

Cost: 13760

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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

Derivation

1. Initial program 33.2%

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

2. Simplified 45.0%

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

   [Start] 33.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 33.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 33.2	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   associate-/r* [=>] 33.2	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
   associate-/r/ [=>] 33.5	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 33.5	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
   times-frac [=>] 34.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
   +-commutative [=>] 34.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   associate--l+ [=>] 42.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   metadata-eval [=>] 42.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   +-rgt-identity [=>] 42.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
   times-frac [=>] 45.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

3. Applied egg-rr 46.3%

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

   [Start] 45.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
   associate-*r* [=>] 46.2	\[ \color{blue}{\left(\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]
   associate-*r/ [=>] 46.1	\[ \color{blue}{\frac{\left(\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right) \cdot \ell}{\tan k}} \]
   div-inv [=>] 45.6	\[ \frac{\left(\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2}}\right)} \cdot \frac{\ell}{\sin k}\right) \cdot \ell}{\tan k} \]
   associate-*l* [=>] 45.1	\[ \frac{\color{blue}{\left(\frac{2}{{t}^{3}} \cdot \left(\frac{1}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\right)} \cdot \ell}{\tan k} \]
   div-inv [=>] 45.1	\[ \frac{\left(\color{blue}{\left(2 \cdot \frac{1}{{t}^{3}}\right)} \cdot \left(\frac{1}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\right) \cdot \ell}{\tan k} \]
   pow-flip [=>] 45.9	\[ \frac{\left(\left(2 \cdot \color{blue}{{t}^{\left(-3\right)}}\right) \cdot \left(\frac{1}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\right) \cdot \ell}{\tan k} \]
   metadata-eval [=>] 45.9	\[ \frac{\left(\left(2 \cdot {t}^{\color{blue}{-3}}\right) \cdot \left(\frac{1}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{\sin k}\right)\right) \cdot \ell}{\tan k} \]
   pow-flip [=>] 46.3	\[ \frac{\left(\left(2 \cdot {t}^{-3}\right) \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{\left(-2\right)}} \cdot \frac{\ell}{\sin k}\right)\right) \cdot \ell}{\tan k} \]
   metadata-eval [=>] 46.3	\[ \frac{\left(\left(2 \cdot {t}^{-3}\right) \cdot \left({\left(\frac{k}{t}\right)}^{\color{blue}{-2}} \cdot \frac{\ell}{\sin k}\right)\right) \cdot \ell}{\tan k} \]

4. Taylor expanded in t around 0 73.0%

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

5. Simplified 73.2%

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

   [Start] 73.0	\[ \frac{2 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\sin k \cdot t\right)}}{\tan k} \]
   associate-*r/ [=>] 73.0	\[ \frac{\color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{2} \cdot \left(\sin k \cdot t\right)}}}{\tan k} \]
   associate-/r* [=>] 73.1	\[ \frac{\color{blue}{\frac{\frac{2 \cdot {\ell}^{2}}{{k}^{2}}}{\sin k \cdot t}}}{\tan k} \]
   unpow2 [=>] 73.1	\[ \frac{\frac{\frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2}}}{\sin k \cdot t}}{\tan k} \]
   associate-*r* [=>] 73.2	\[ \frac{\frac{\frac{\color{blue}{\left(2 \cdot \ell\right) \cdot \ell}}{{k}^{2}}}{\sin k \cdot t}}{\tan k} \]
   unpow2 [=>] 73.2	\[ \frac{\frac{\frac{\left(2 \cdot \ell\right) \cdot \ell}{\color{blue}{k \cdot k}}}{\sin k \cdot t}}{\tan k} \]

6. Applied egg-rr 99.2%

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

   [Start] 73.2	\[ \frac{\frac{\frac{\left(2 \cdot \ell\right) \cdot \ell}{k \cdot k}}{\sin k \cdot t}}{\tan k} \]
   times-frac [=>] 92.2	\[ \frac{\frac{\color{blue}{\frac{2 \cdot \ell}{k} \cdot \frac{\ell}{k}}}{\sin k \cdot t}}{\tan k} \]
   times-frac [=>] 99.2	\[ \frac{\color{blue}{\frac{\frac{2 \cdot \ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}}}{\tan k} \]
   *-un-lft-identity [=>] 99.2	\[ \frac{\frac{\frac{2 \cdot \ell}{\color{blue}{1 \cdot k}}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}}{\tan k} \]
   times-frac [=>] 99.2	\[ \frac{\frac{\color{blue}{\frac{2}{1} \cdot \frac{\ell}{k}}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}}{\tan k} \]
   metadata-eval [=>] 99.2	\[ \frac{\frac{\color{blue}{2} \cdot \frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{t}}{\tan k} \]

7. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	93.8%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.18 \cdot 10^{-129} \lor \neg \left(k \leq 9.4 \cdot 10^{-105}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k \cdot t}}{\sin k} \cdot \frac{\frac{2}{k}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \end{array} \]

Alternative 2
Accuracy	94.3%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+212}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell}{\sin k}}{\frac{k \cdot t}{\frac{\ell}{\tan k}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\ell}{k \cdot t}}{\sin k} \cdot \frac{\frac{2}{k}}{\tan k}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	13760

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

Alternative 4
Accuracy	72.8%
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	7360

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

Alternative 6
Accuracy	70.0%
Cost	960

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

Alternative 7
Accuracy	70.3%
Cost	960

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

Alternative 8
Accuracy	70.4%
Cost	960

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

Alternative 9
Accuracy	72.3%
Cost	960

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

Alternative 10
Accuracy	33.3%
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \]

Reproduce

herbie shell --seed 2023160 
(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))))