Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 1.5

Time: 31.8s

Precision: binary64

Cost: 13760

Math

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

↓

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

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 (* (tan k) (* (/ k (/ l (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 2.0 / (tan(k) * ((k / (l / 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 = 2.0d0 / (tan(k) * ((k / (l / 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 2.0 / (Math.tan(k) * ((k / (l / 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 2.0 / (math.tan(k) * ((k / (l / 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(2.0 / Float64(tan(k) * Float64(Float64(k / Float64(l / sin(k))) * Float64(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 = 2.0 / (tan(k) * ((k / (l / 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[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $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 39.9

   \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
   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+ [=>] 39.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
   metadata-eval [=>] 39.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

3. Taylor expanded in t around 0 22.0

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

4. Simplified 22.1

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

   [Start] 22.0	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
   associate-/l* [=>] 22.1	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{\sin k \cdot t}}}} \]
   unpow2 [=>] 22.1	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{\sin k \cdot t}}} \]
   unpow2 [=>] 22.1	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot t}}} \]

5. Applied egg-rr 6.6

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

6. Applied egg-rr 1.5

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

7. Final simplification 1.5

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

Alternatives

Alternative 1
Error	11.7
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -6.6 \cdot 10^{-39} \lor \neg \left(k \leq 4.2 \cdot 10^{-96}\right):\\ \;\;\;\;2 \cdot \frac{\ell}{\tan k \cdot \left(\sin k \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{\ell}{k}}\right)\right)}\\ \end{array} \]

Alternative 2
Error	5.1
Cost	14025

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-193} \lor \neg \left(\ell \leq 2.7 \cdot 10^{-238}\right):\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 3
Error	5.1
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq 8.2 \cdot 10^{-110} \lor \neg \left(t \leq 3.5 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(k \cdot \left(\frac{\sin k}{\ell} \cdot \left(k \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	4.7
Cost	13892

\[\begin{array}{l} \mathbf{if}\;t \leq 1.28 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\sin k \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	13892

\[\begin{array}{l} \mathbf{if}\;t \leq 1.28 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \frac{\ell}{\sin k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	22.8
Cost	7880

\[\begin{array}{l} t_1 := \left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot t_1}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot \left(-\frac{k}{\ell}\right)\right)\right)}{\frac{\cos k}{-k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \end{array} \]

Alternative 7
Error	22.5
Cost	1736

\[\begin{array}{l} t_1 := \left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-106}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot t_1}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} + \frac{\ell}{t} \cdot \left(\ell \cdot -0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \end{array} \]

Alternative 8
Error	22.7
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-209} \lor \neg \left(t \leq 0.0072\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t}}{\left(k \cdot k\right) \cdot \frac{k \cdot k}{\ell}}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-209} \lor \neg \left(t \leq 9.6 \cdot 10^{-155}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{k}}{k}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -1.52 \cdot 10^{-209} \lor \neg \left(t \leq 0.002\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}}\\ \end{array} \]

Alternative 11
Error	22.8
Cost	1224

\[\begin{array}{l} t_1 := \left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-208}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot t_1}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \end{array} \]

Alternative 12
Error	25.5
Cost	960

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

Alternative 13
Error	25.1
Cost	960

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

Alternative 14
Error	24.3
Cost	960

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

Alternative 15
Error	23.1
Cost	960

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

Reproduce

herbie shell --seed 2023083 
(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))))