Toniolo and Linder, Equation (10-)

Average Error: 73.92% → 1.17%

Time: 31.7s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-71} \lor \neg \left(k \leq 8.4 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{2}{t \cdot \frac{k \cdot {\sin k}^{2}}{\ell}} \cdot \frac{\ell}{\frac{k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \end{array} \]

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
 (if (or (<= k -3.4e-71) (not (<= k 8.4e-50)))
   (* (/ 2.0 (* t (/ (* k (pow (sin k) 2.0)) l))) (/ l (/ k (cos k))))
   (* (/ (/ (/ l k) (* k (- t))) (* k (/ k l))) -2.0)))

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) {
	double tmp;
	if ((k <= -3.4e-71) || !(k <= 8.4e-50)) {
		tmp = (2.0 / (t * ((k * pow(sin(k), 2.0)) / l))) * (l / (k / cos(k)));
	} else {
		tmp = (((l / k) / (k * -t)) / (k * (k / l))) * -2.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((k <= (-3.4d-71)) .or. (.not. (k <= 8.4d-50))) then
        tmp = (2.0d0 / (t * ((k * (sin(k) ** 2.0d0)) / l))) * (l / (k / cos(k)))
    else
        tmp = (((l / k) / (k * -t)) / (k * (k / l))) * (-2.0d0)
    end if
    code = tmp
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) {
	double tmp;
	if ((k <= -3.4e-71) || !(k <= 8.4e-50)) {
		tmp = (2.0 / (t * ((k * Math.pow(Math.sin(k), 2.0)) / l))) * (l / (k / Math.cos(k)));
	} else {
		tmp = (((l / k) / (k * -t)) / (k * (k / l))) * -2.0;
	}
	return tmp;
}

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):
	tmp = 0
	if (k <= -3.4e-71) or not (k <= 8.4e-50):
		tmp = (2.0 / (t * ((k * math.pow(math.sin(k), 2.0)) / l))) * (l / (k / math.cos(k)))
	else:
		tmp = (((l / k) / (k * -t)) / (k * (k / l))) * -2.0
	return tmp

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)
	tmp = 0.0
	if ((k <= -3.4e-71) || !(k <= 8.4e-50))
		tmp = Float64(Float64(2.0 / Float64(t * Float64(Float64(k * (sin(k) ^ 2.0)) / l))) * Float64(l / Float64(k / cos(k))));
	else
		tmp = Float64(Float64(Float64(Float64(l / k) / Float64(k * Float64(-t))) / Float64(k * Float64(k / l))) * -2.0);
	end
	return tmp
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_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -3.4e-71) || ~((k <= 8.4e-50)))
		tmp = (2.0 / (t * ((k * (sin(k) ^ 2.0)) / l))) * (l / (k / cos(k)));
	else
		tmp = (((l / k) / (k * -t)) / (k * (k / l))) * -2.0;
	end
	tmp_2 = tmp;
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_] := If[Or[LessEqual[k, -3.4e-71], N[Not[LessEqual[k, 8.4e-50]], $MachinePrecision]], N[(N[(2.0 / N[(t * N[(N[(k * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(l / k), $MachinePrecision] / N[(k * (-t)), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -3.40000000000000003e-71 or 8.4000000000000003e-50 < k

   1. Initial program 70.37

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

   2. Taylor expanded in t around 0 29.85

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

   3. Simplified 23.04

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

      [Start] 29.85	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 30.02	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 30.02	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 30.02	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 30.02	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 23.04	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   4. Applied egg-rr 14.12

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

   5. Simplified 1.1

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

      [Start] 14.12	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 1.13	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]
      associate-/r/ [=>] 1.1	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}} \]

   6. Applied egg-rr 0.62

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

   if -3.40000000000000003e-71 < k < 8.4000000000000003e-50

   1. Initial program 98.98

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

   2. Simplified 81.56

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

      [Start] 98.98	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 98.93	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 98.94	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 98.94	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 97.82	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 97.82	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 97.82	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 98.38	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 98.38	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 98.38	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 87.36

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 85.03

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 87.36	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 87.36	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 85.03	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 34.67

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

   6. Taylor expanded in l around 0 34.67

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

   7. Simplified 25.07

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

      [Start] 34.67	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \]
      unpow2 [=>] 34.67	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]
      associate-*r* [<=] 34.52	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]
      associate-/r* [=>] 25.07	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot t}}\right) \]

   8. Applied egg-rr 5

      \[\leadsto 2 \cdot \color{blue}{\left(-\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{\frac{k}{\ell} \cdot k}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.17

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-71} \lor \neg \left(k \leq 8.4 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{2}{t \cdot \frac{k \cdot {\sin k}^{2}}{\ell}} \cdot \frac{\ell}{\frac{k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \end{array} \]

Alternatives

Alternative 1
Error	12.25%
Cost	14540

\[\begin{array}{l} t_1 := \frac{2}{\frac{k \cdot t}{\frac{\ell}{0.5 - \frac{\cos \left(k + k\right)}{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{if}\;k \leq -0.033:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+179}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	2.02%
Cost	14473

\[\begin{array}{l} \mathbf{if}\;k \leq -0.000105 \lor \neg \left(k \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\cos k}{k} \cdot \left(-\ell\right)} \cdot \left(k \cdot \frac{\cos \left(k + k\right) + -1}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \end{array} \]

Alternative 3
Error	20.74%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-10} \lor \neg \left(k \leq 3.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t \cdot \sin k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \end{array} \]

Alternative 4
Error	33.95%
Cost	13960

\[\begin{array}{l} \mathbf{if}\;t \leq -102000000:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \end{array} \]

Alternative 5
Error	35.07%
Cost	8200

\[\begin{array}{l} \mathbf{if}\;t \leq -0.00014:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \left(-t\right)}}{k \cdot \frac{k}{\ell}} \cdot -2\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \frac{t}{\left(\frac{\ell}{k} \cdot -0.5 + \frac{\ell}{{k}^{3}}\right) + \frac{\ell}{k} \cdot 0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{1}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \end{array} \]

Alternative 6
Error	35.35%
Cost	1024

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

Alternative 7
Error	39.35%
Cost	960

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

Alternative 8
Error	37.83%
Cost	960

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

Alternative 9
Error	37.44%
Cost	960

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

Alternative 10
Error	35.77%
Cost	960

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

Reproduce

herbie shell --seed 2023115 
(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))))