Toniolo and Linder, Equation (10+)

Average Error: 31.9 → 11.8

Time: 15.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\\ t_2 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_1\\ t_3 := \frac{2}{\left(\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell} \cdot \tan k\right) \cdot t_1}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-257}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot k}{\ell}}{\ell \cdot \frac{\frac{\cos k}{k}}{t_4}}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t_4 + 0\right) \cdot \frac{t \cdot \frac{k}{\ell}}{\frac{\ell \cdot \cos k}{k}}}\\ \end{array} \]

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 (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))
        (t_2 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_1))
        (t_3 (/ 2.0 (* (* (/ (* (sin k) (/ (pow t 3.0) l)) l) (tan k)) t_1)))
        (t_4 (pow (sin k) 2.0)))
   (if (<= t_2 -2e-257)
     t_3
     (if (<= t_2 0.0)
       (/ 2.0 (/ (/ (* t k) l) (* l (/ (/ (cos k) k) t_4))))
       (if (<= t_2 INFINITY)
         t_3
         (/ 2.0 (* (+ t_4 0.0) (/ (* t (/ k l)) (/ (* l (cos k)) k)))))))))

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 = (1.0 + pow((k / t), 2.0)) + 1.0;
	double t_2 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1;
	double t_3 = 2.0 / ((((sin(k) * (pow(t, 3.0) / l)) / l) * tan(k)) * t_1);
	double t_4 = pow(sin(k), 2.0);
	double tmp;
	if (t_2 <= -2e-257) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / (((t * k) / l) / (l * ((cos(k) / k) / t_4)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = 2.0 / ((t_4 + 0.0) * ((t * (k / l)) / ((l * cos(k)) / k)));
	}
	return tmp;
}

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 = (1.0 + Math.pow((k / t), 2.0)) + 1.0;
	double t_2 = (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_1;
	double t_3 = 2.0 / ((((Math.sin(k) * (Math.pow(t, 3.0) / l)) / l) * Math.tan(k)) * t_1);
	double t_4 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_2 <= -2e-257) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = 2.0 / (((t * k) / l) / (l * ((Math.cos(k) / k) / t_4)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = 2.0 / ((t_4 + 0.0) * ((t * (k / l)) / ((l * Math.cos(k)) / k)));
	}
	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 = (1.0 + math.pow((k / t), 2.0)) + 1.0
	t_2 = (((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * t_1
	t_3 = 2.0 / ((((math.sin(k) * (math.pow(t, 3.0) / l)) / l) * math.tan(k)) * t_1)
	t_4 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_2 <= -2e-257:
		tmp = t_3
	elif t_2 <= 0.0:
		tmp = 2.0 / (((t * k) / l) / (l * ((math.cos(k) / k) / t_4)))
	elif t_2 <= math.inf:
		tmp = t_3
	else:
		tmp = 2.0 / ((t_4 + 0.0) * ((t * (k / l)) / ((l * math.cos(k)) / k)))
	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(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)
	t_2 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_1)
	t_3 = Float64(2.0 / Float64(Float64(Float64(Float64(sin(k) * Float64((t ^ 3.0) / l)) / l) * tan(k)) * t_1))
	t_4 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_2 <= -2e-257)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * k) / l) / Float64(l * Float64(Float64(cos(k) / k) / t_4))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(2.0 / Float64(Float64(t_4 + 0.0) * Float64(Float64(t * Float64(k / l)) / Float64(Float64(l * cos(k)) / k))));
	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 = (1.0 + ((k / t) ^ 2.0)) + 1.0;
	t_2 = ((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * t_1;
	t_3 = 2.0 / ((((sin(k) * ((t ^ 3.0) / l)) / l) * tan(k)) * t_1);
	t_4 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_2 <= -2e-257)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = 2.0 / (((t * k) / l) / (l * ((cos(k) / k) / t_4)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = 2.0 / ((t_4 + 0.0) * ((t * (k / l)) / ((l * cos(k)) / k)));
	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[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = 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] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(N[(N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$2, -2e-257], t$95$3, If[LessEqual[t$95$2, 0.0], N[(2.0 / N[(N[(N[(t * k), $MachinePrecision] / l), $MachinePrecision] / N[(l * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(2.0 / N[(N[(t$95$4 + 0.0), $MachinePrecision] * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < -2e-257 or 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 12.5

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

   2. Applied egg-rr 9.8

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

   if -2e-257 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 0.0

   1. Initial program 57.2

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

   2. Taylor expanded in t around 0 47.9

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

   3. Simplified 42.9

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

   4. Applied egg-rr 41.0

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

   5. Applied egg-rr 17.7

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

   6. Applied egg-rr 18.6

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   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. Taylor expanded in t around 0 40.3

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

   3. Simplified 40.0

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

   4. Applied egg-rr 34.0

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

   5. Applied egg-rr 14.8

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

   6. Applied egg-rr 14.8

      \[\leadsto \frac{2}{\color{blue}{\left({\sin k}^{2} + 0\right)} \cdot \frac{t \cdot \frac{k}{\ell}}{\frac{\ell \cdot \cos k}{k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot k}{\ell}}{\ell \cdot \frac{\frac{\cos k}{k}}{{\sin k}^{2}}}}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty:\\ \;\;\;\;\frac{2}{\left(\frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} + 0\right) \cdot \frac{t \cdot \frac{k}{\ell}}{\frac{\ell \cdot \cos k}{k}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(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))))