Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 21.8

Time: 30.4s

Precision: binary64

Cost: 26432

Math

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

↓

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

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
 (/ 2.0 (* (/ (* t (pow k 2.0)) (pow l 2.0)) (* (sin k) (tan 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) {
	return 2.0 / (((t * pow(k, 2.0)) / pow(l, 2.0)) * (sin(k) * tan(k)));
}

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
    code = 2.0d0 / (((t * (k ** 2.0d0)) / (l ** 2.0d0)) * (sin(k) * tan(k)))
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) {
	return 2.0 / (((t * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.sin(k) * Math.tan(k)));
}

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):
	return 2.0 / (((t * math.pow(k, 2.0)) / math.pow(l, 2.0)) * (math.sin(k) * math.tan(k)))

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)
	return Float64(2.0 / Float64(Float64(Float64(t * (k ^ 2.0)) / (l ^ 2.0)) * Float64(sin(k) * tan(k))))
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 = code(t, l, k)
	tmp = 2.0 / (((t * (k ^ 2.0)) / (l ^ 2.0)) * (sin(k) * tan(k)));
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_] := N[(2.0 / N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.7

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

2. Simplified 40.2

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

   [Start] 47.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 47.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 47.7	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 47.6	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 47.6	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-107 [=>] 40.2	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)} \cdot \left(\tan k \cdot \sin k\right)\right)} \]
   metadata-eval [=>] 40.2	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right) \cdot \left(\tan k \cdot \sin k\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 40.2	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\tan k \cdot \sin k\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.2	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}\right)} \]

3. Applied egg-rr 39.9

   \[\leadsto \frac{2}{\color{blue}{0 - {\left(\frac{k}{t}\right)}^{2} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)}} \]

4. Simplified 39.9

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

   [Start] 39.9	\[ \frac{2}{0 - {\left(\frac{k}{t}\right)}^{2} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(-\tan k\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 39.9	\[ \frac{2}{0 - {\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\left(\sin k \cdot \left(-\tan k\right)\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 39.9	\[ \frac{2}{0 - \color{blue}{\left(\sin k \cdot \left(-\tan k\right)\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 39.9	\[ \frac{2}{0 - \left(\sin k \cdot \left(-\tan k\right)\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 39.9	\[ \frac{2}{0 - \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(-\tan k\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-38 [<=] 60.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot 0} - \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(-\tan k\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 60.1	\[ \frac{2}{\color{blue}{0 \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right)} - \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(-\tan k\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 39.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(0 - \sin k \cdot \left(-\tan k\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-5 [=>] 39.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(-\sin k \cdot \left(-\tan k\right)\right)}} \]

5. Taylor expanded in t around 0 21.8

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

6. Applied egg-rr 21.8

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right) + 0}} \]

7. Simplified 21.8

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

   [Start] 21.8	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right) + 0} \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 21.8	\[ \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 21.8	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

8. Final simplification 21.8

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

Alternatives

Alternative 1
Error	21.8
Cost	26432

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

Alternative 2
Error	31.5
Cost	19904

\[2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{4} \cdot t} \]

Alternative 3
Error	30.6
Cost	19904

\[\frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{{\ell}^{2}}} \]

Alternative 4
Error	31.6
Cost	13376

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

Reproduce

herbie shell --seed 2023090 
(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))))