Toniolo and Linder, Equation (10-)

Average Accuracy: 25.8% → 98.8%

Time: 36.8s

Precision: binary64

Cost: 13760

Math

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

↓

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

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

Derivation

1. Initial program 25.8%

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

2. Simplified 37.6%

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

   [Start] 25.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   *-commutative [=>] 25.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   associate-*l* [=>] 25.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
   +-commutative [=>] 25.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
   associate--l+ [=>] 37.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 37.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 65.5%

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

4. Applied egg-rr 66.0%

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

5. Simplified 97.4%

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

   [Start] 66.0	\[ \frac{2}{\tan k \cdot \left(\frac{k \cdot k}{\ell \cdot \left(-\ell\right)} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)} \]
   times-frac [=>] 88.8	\[ \frac{2}{\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{-\ell}\right)} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)} \]
   associate-*l* [=>] 97.4	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \left(\frac{k}{-\ell} \cdot \left(\sin k \cdot \left(-t\right)\right)\right)\right)}} \]

6. Applied egg-rr 98.2%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternatives

Alternative 1
Accuracy	91.4%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-53} \lor \neg \left(k \leq 3.6 \cdot 10^{-13}\right):\\ \;\;\;\;\ell \cdot \left(\frac{\frac{2}{k}}{\tan k} \cdot \frac{\ell}{k \cdot \left(\sin k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(k \cdot \frac{t}{\frac{\frac{\ell}{k}}{k}}\right)}\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	13760

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

Alternative 3
Accuracy	63.3%
Cost	7360

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

Alternative 4
Accuracy	63.5%
Cost	1088

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

Alternative 5
Accuracy	59.8%
Cost	960

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

Alternative 6
Accuracy	61.0%
Cost	960

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

Alternative 7
Accuracy	60.2%
Cost	960

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

Alternative 8
Accuracy	60.0%
Cost	960

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

Alternative 9
Accuracy	61.4%
Cost	960

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

Alternative 10
Accuracy	63.6%
Cost	960

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

Reproduce

herbie shell --seed 2023129 
(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))))