Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 4.8

Time: 24.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sin k \cdot \sqrt{t}\\ \mathbf{if}\;t \leq 6.915526448265006 \cdot 10^{-266}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{{\left(\frac{k}{\ell}\right)}^{2}}{\cos k} \cdot t_1\right)}\\ \end{array} \]

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 (* (sin k) (sqrt t))))
   (if (<= t 6.915526448265006e-266)
     (/ 2.0 (* (/ k l) (* (/ k l) (/ (* t (pow (sin k) 2.0)) (cos k)))))
     (/ 2.0 (* t_1 (* (/ (pow (/ k l) 2.0) (cos 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 = sin(k) * sqrt(t);
	double tmp;
	if (t <= 6.915526448265006e-266) {
		tmp = 2.0 / ((k / l) * ((k / l) * ((t * pow(sin(k), 2.0)) / cos(k))));
	} else {
		tmp = 2.0 / (t_1 * ((pow((k / l), 2.0) / cos(k)) * 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 = sin(k) * sqrt(t)
    if (t <= 6.915526448265006d-266) then
        tmp = 2.0d0 / ((k / l) * ((k / l) * ((t * (sin(k) ** 2.0d0)) / cos(k))))
    else
        tmp = 2.0d0 / (t_1 * ((((k / l) ** 2.0d0) / cos(k)) * 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 = Math.sin(k) * Math.sqrt(t);
	double tmp;
	if (t <= 6.915526448265006e-266) {
		tmp = 2.0 / ((k / l) * ((k / l) * ((t * Math.pow(Math.sin(k), 2.0)) / Math.cos(k))));
	} else {
		tmp = 2.0 / (t_1 * ((Math.pow((k / l), 2.0) / Math.cos(k)) * 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 = math.sin(k) * math.sqrt(t)
	tmp = 0
	if t <= 6.915526448265006e-266:
		tmp = 2.0 / ((k / l) * ((k / l) * ((t * math.pow(math.sin(k), 2.0)) / math.cos(k))))
	else:
		tmp = 2.0 / (t_1 * ((math.pow((k / l), 2.0) / math.cos(k)) * 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(sin(k) * sqrt(t))
	tmp = 0.0
	if (t <= 6.915526448265006e-266)
		tmp = Float64(2.0 / Float64(Float64(k / l) * Float64(Float64(k / l) * Float64(Float64(t * (sin(k) ^ 2.0)) / cos(k)))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64((Float64(k / l) ^ 2.0) / cos(k)) * 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 = sin(k) * sqrt(t);
	tmp = 0.0;
	if (t <= 6.915526448265006e-266)
		tmp = 2.0 / ((k / l) * ((k / l) * ((t * (sin(k) ^ 2.0)) / cos(k))));
	else
		tmp = 2.0 / (t_1 * ((((k / l) ^ 2.0) / cos(k)) * 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[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.915526448265006e-266], N[(2.0 / N[(N[(k / l), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < 6.915526448265006e-266

   1. Initial program 49.4

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

   2. Simplified 42.0

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

   3. Taylor expanded in t around 0 23.5

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

   4. Applied egg-rr 23.2

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

   5. Applied egg-rr 11.1

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

   6. Applied egg-rr 4.2

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

   if 6.915526448265006e-266 < t

   1. Initial program 46.9

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

   2. Simplified 38.6

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

   3. Taylor expanded in t around 0 22.1

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

   4. Applied egg-rr 20.5

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

   5. Applied egg-rr 8.6

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

   6. Applied egg-rr 5.4

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

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.915526448265006 \cdot 10^{-266}:\\ \;\;\;\;\frac{2}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \sqrt{t}\right) \cdot \left(\frac{{\left(\frac{k}{\ell}\right)}^{2}}{\cos k} \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))