Toniolo and Linder, Equation (10-)

Average Error: 47.5 → 1.1

Time: 26.8s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{\ell}{k \cdot \sin k}}{t \cdot \frac{k}{\ell}}}{\tan k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-145}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{\sin k}}{\frac{k}{\ell} \cdot \left(k \cdot t\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_1 (/ (* 2.0 (/ (/ l (* k (sin k))) (* t (/ k l)))) (tan k))))
   (if (<= k -1e-150)
     t_1
     (if (<= k 1e-145)
       (/ (* 2.0 (/ (/ l (sin k)) (* (/ k l) (* k t)))) (tan k))
       t_1))))

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 t_1 = (2.0 * ((l / (k * sin(k))) / (t * (k / l)))) / tan(k);
	double tmp;
	if (k <= -1e-150) {
		tmp = t_1;
	} else if (k <= 1e-145) {
		tmp = (2.0 * ((l / sin(k)) / ((k / l) * (k * t)))) / tan(k);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 * ((l / (k * sin(k))) / (t * (k / l)))) / tan(k)
    if (k <= (-1d-150)) then
        tmp = t_1
    else if (k <= 1d-145) then
        tmp = (2.0d0 * ((l / sin(k)) / ((k / l) * (k * t)))) / tan(k)
    else
        tmp = t_1
    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 t_1 = (2.0 * ((l / (k * Math.sin(k))) / (t * (k / l)))) / Math.tan(k);
	double tmp;
	if (k <= -1e-150) {
		tmp = t_1;
	} else if (k <= 1e-145) {
		tmp = (2.0 * ((l / Math.sin(k)) / ((k / l) * (k * t)))) / Math.tan(k);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = (2.0 * ((l / (k * math.sin(k))) / (t * (k / l)))) / math.tan(k)
	tmp = 0
	if k <= -1e-150:
		tmp = t_1
	elif k <= 1e-145:
		tmp = (2.0 * ((l / math.sin(k)) / ((k / l) * (k * t)))) / math.tan(k)
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(2.0 * Float64(Float64(l / Float64(k * sin(k))) / Float64(t * Float64(k / l)))) / tan(k))
	tmp = 0.0
	if (k <= -1e-150)
		tmp = t_1;
	elseif (k <= 1e-145)
		tmp = Float64(Float64(2.0 * Float64(Float64(l / sin(k)) / Float64(Float64(k / l) * Float64(k * t)))) / tan(k));
	else
		tmp = t_1;
	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)
	t_1 = (2.0 * ((l / (k * sin(k))) / (t * (k / l)))) / tan(k);
	tmp = 0.0;
	if (k <= -1e-150)
		tmp = t_1;
	elseif (k <= 1e-145)
		tmp = (2.0 * ((l / sin(k)) / ((k / l) * (k * t)))) / tan(k);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(2.0 * N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1e-150], t$95$1, If[LessEqual[k, 1e-145], N[(N[(2.0 * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.00000000000000001e-150 or 9.99999999999999915e-146 < k

   1. Initial program 47.0

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

   2. Simplified 36.2

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

   3. Applied egg-rr 36.1

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

   4. Taylor expanded in l around 0 20.5

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

   5. Simplified 6.6

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

   6. Taylor expanded in l around 0 20.5

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

   7. Simplified 4.6

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

   8. Applied egg-rr 1.1

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

   if -1.00000000000000001e-150 < k < 9.99999999999999915e-146

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Applied egg-rr 59.2

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

   4. Taylor expanded in l around 0 60.7

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

   5. Simplified 18.7

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

   6. Taylor expanded in l around 0 60.7

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

   7. Simplified 2.6

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

   8. Applied egg-rr 2.5

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k \cdot \sin k}}{t \cdot \frac{k}{\ell}}}{\tan k}\\ \mathbf{elif}\;k \leq 10^{-145}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{\sin k}}{\frac{k}{\ell} \cdot \left(k \cdot t\right)}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\ell}{k \cdot \sin k}}{t \cdot \frac{k}{\ell}}}{\tan k}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.9
Cost	14280

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{\ell}{\sin k}}{\frac{k}{\ell} \cdot \left(k \cdot t\right)}}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell}}}{t}}{k}}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.6
Cost	14156

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell}}}{t}}{k}}{\tan k}\\ \mathbf{if}\;k \leq -100:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-62}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} + 0.16666666666666666 \cdot \left(\ell \cdot k\right)}{k \cdot t}\right)}{\tan k}\\ \mathbf{elif}\;k \leq 5.16456434438668 \cdot 10^{+152}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\frac{\ell}{k \cdot k}}{\sin k} \cdot \frac{\ell}{t}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.9
Cost	14024

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot \sin k}}{k \cdot t}}{\tan k}\\ \mathbf{if}\;k \leq -100:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} + 0.16666666666666666 \cdot \left(\ell \cdot k\right)}{k \cdot t}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.0
Cost	14024

\[\begin{array}{l} t_1 := \frac{2 \cdot \frac{\frac{\frac{\frac{\ell}{k}}{\frac{\sin k}{\ell}}}{t}}{k}}{\tan k}\\ \mathbf{if}\;k \leq -100:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k} + 0.16666666666666666 \cdot \left(\ell \cdot k\right)}{k \cdot t}\right)}{\tan k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	23.2
Cost	13960

\[\begin{array}{l} t_1 := \frac{2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)}{\tan k}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 243956695372.8898:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{2}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.4
Cost	13760

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

Alternative 7
Error	23.8
Cost	7360

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

Alternative 8
Error	24.1
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq 10^{-150}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)}}{k}}{k}\\ \mathbf{elif}\;k \leq 10^{-90}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\frac{\ell}{k}}{t}}{k} \cdot \left(\frac{2}{k \cdot k} + 0.6666666666666666\right)\right)\\ \end{array} \]

Alternative 9
Error	24.4
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{\frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)}}{k}}{k}\\ t_2 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-196}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot t_2}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.6
Cost	1088

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ 2 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{1}{t}\right)\right) \end{array} \]

Alternative 11
Error	26.0
Cost	960

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

Alternative 12
Error	25.1
Cost	960

\[\frac{\ell \cdot \frac{\frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot 0.5\right)}}{k}}{k} \]

Alternative 13
Error	24.6
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ 2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right) \end{array} \]

Reproduce

herbie shell --seed 2022217 
(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))))