Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 17.1

Time: 26.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{{\left(\sqrt{t}\right)}^{3}}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 5.78035664198339 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k \cdot {\ell}^{2}}{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t_1 \cdot \left(\sin k \cdot t_1\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

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
 (let* ((t_1 (/ (pow (sqrt t) 3.0) l)))
   (if (<= (* l l) 5.78035664198339e+298)
     (/ 2.0 (/ k (/ (* (cos k) (pow l 2.0)) (* k (* (sin k) (* t (sin k)))))))
     (/ 2.0 (* (* (* t_1 (* (sin k) t_1)) (tan k)) (pow (/ k t) 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 t_1 = pow(sqrt(t), 3.0) / l;
	double tmp;
	if ((l * l) <= 5.78035664198339e+298) {
		tmp = 2.0 / (k / ((cos(k) * pow(l, 2.0)) / (k * (sin(k) * (t * sin(k))))));
	} else {
		tmp = 2.0 / (((t_1 * (sin(k) * t_1)) * tan(k)) * pow((k / t), 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) :: t_1
    real(8) :: tmp
    t_1 = (sqrt(t) ** 3.0d0) / l
    if ((l * l) <= 5.78035664198339d+298) then
        tmp = 2.0d0 / (k / ((cos(k) * (l ** 2.0d0)) / (k * (sin(k) * (t * sin(k))))))
    else
        tmp = 2.0d0 / (((t_1 * (sin(k) * t_1)) * tan(k)) * ((k / t) ** 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 t_1 = Math.pow(Math.sqrt(t), 3.0) / l;
	double tmp;
	if ((l * l) <= 5.78035664198339e+298) {
		tmp = 2.0 / (k / ((Math.cos(k) * Math.pow(l, 2.0)) / (k * (Math.sin(k) * (t * Math.sin(k))))));
	} else {
		tmp = 2.0 / (((t_1 * (Math.sin(k) * t_1)) * Math.tan(k)) * Math.pow((k / t), 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):
	t_1 = math.pow(math.sqrt(t), 3.0) / l
	tmp = 0
	if (l * l) <= 5.78035664198339e+298:
		tmp = 2.0 / (k / ((math.cos(k) * math.pow(l, 2.0)) / (k * (math.sin(k) * (t * math.sin(k))))))
	else:
		tmp = 2.0 / (((t_1 * (math.sin(k) * t_1)) * math.tan(k)) * math.pow((k / t), 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)
	t_1 = Float64((sqrt(t) ^ 3.0) / l)
	tmp = 0.0
	if (Float64(l * l) <= 5.78035664198339e+298)
		tmp = Float64(2.0 / Float64(k / Float64(Float64(cos(k) * (l ^ 2.0)) / Float64(k * Float64(sin(k) * Float64(t * sin(k)))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_1 * Float64(sin(k) * t_1)) * tan(k)) * (Float64(k / t) ^ 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)
	t_1 = (sqrt(t) ^ 3.0) / l;
	tmp = 0.0;
	if ((l * l) <= 5.78035664198339e+298)
		tmp = 2.0 / (k / ((cos(k) * (l ^ 2.0)) / (k * (sin(k) * (t * sin(k))))));
	else
		tmp = 2.0 / (((t_1 * (sin(k) * t_1)) * tan(k)) * ((k / t) ^ 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_] := Block[{t$95$1 = N[(N[Power[N[Sqrt[t], $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 5.78035664198339e+298], N[(2.0 / N[(k / N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(k * N[(N[Sin[k], $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $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 (*.f64 l l) < 5.7803566419833897e298

   1. Initial program 45.2

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

   2. Simplified 35.9

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

   3. Taylor expanded in t around 0 15.2

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

   4. Applied unpow2_binary64 15.2

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

   5. Applied associate-*l*_binary64 12.9

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

   6. Applied associate-/l*_binary64 10.4

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

   7. Applied unpow2_binary64 10.4

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

   8. Applied associate-*r*_binary64 10.2

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

   if 5.7803566419833897e298 < (*.f64 l l)

   1. Initial program 63.8

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

   2. Simplified 63.7

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

   3. Applied add-sqr-sqrt_binary64 63.7

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

   4. Applied unpow-prod-down_binary64 63.7

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

   5. Applied times-frac_binary64 53.6

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

   6. Applied associate-*l*_binary64 53.6

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.78035664198339 \cdot 10^{+298}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k \cdot {\ell}^{2}}{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt{t}\right)}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{{\left(\sqrt{t}\right)}^{3}}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(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))))