Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 0.7

Time: 28.8s

Precision: binary64

Cost: 20288

Math

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

↓

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

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 (* (/ (sin k) (/ l k)) (/ (/ t (cos k)) (/ (/ l k) (sin 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 / ((sin(k) / (l / k)) * ((t / cos(k)) / ((l / k) / sin(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 / ((sin(k) / (l / k)) * ((t / cos(k)) / ((l / k) / sin(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 / ((Math.sin(k) / (l / k)) * ((t / Math.cos(k)) / ((l / k) / Math.sin(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 / ((math.sin(k) / (l / k)) * ((t / math.cos(k)) / ((l / k) / math.sin(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(sin(k) / Float64(l / k)) * Float64(Float64(t / cos(k)) / Float64(Float64(l / k) / sin(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 / ((sin(k) / (l / k)) * ((t / cos(k)) / ((l / k) / sin(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[Sin[k], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.8

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

2. Taylor expanded in t around 0 22.6

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

3. Simplified 16.0

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{{\sin k}^{2}}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{t}{\cos k}}} \]
   Proof
   (*.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (*.f64 k k) l)) (/.f64 t (cos.f64 k))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 (pow.f64 (sin.f64 k) 2) l) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) l)) (/.f64 t (cos.f64 k))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 k 2)) (*.f64 l l))) (/.f64 t (cos.f64 k))): 55 points increase in error, 8 points decrease in error
   (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2))) (*.f64 l l)) (/.f64 t (cos.f64 k))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (/.f64 t (cos.f64 k))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 k 2) (pow.f64 (sin.f64 k) 2)) t) (*.f64 (pow.f64 l 2) (cos.f64 k)))): 22 points increase in error, 16 points decrease in error
   (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t))) (*.f64 (pow.f64 l 2) (cos.f64 k))): 3 points increase in error, 15 points decrease in error
   (/.f64 (*.f64 (pow.f64 k 2) (*.f64 (pow.f64 (sin.f64 k) 2) t)) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 24.0

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

5. Applied egg-rr 8.7

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

6. Applied egg-rr 0.7

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

7. Final simplification 0.7

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

Alternatives

Alternative 1
Error	3.1
Cost	20224

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

Alternative 2
Error	3.5
Cost	14408

\[\begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{2}{\frac{t_1 \cdot \frac{0.5 + \cos \left(2 \cdot k\right) \cdot -0.5}{\frac{\ell}{k}}}{\frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 10^{-5}:\\ \;\;\;\;\frac{2}{\frac{t_1 \cdot \frac{k \cdot k}{\frac{\ell}{k}}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	25.2
Cost	7488

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

Alternative 4
Error	30.3
Cost	960

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

Alternative 5
Error	28.6
Cost	960

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

Alternative 6
Error	27.4
Cost	960

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

Alternative 7
Error	26.5
Cost	960

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

Alternative 8
Error	26.5
Cost	960

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

Alternative 9
Error	26.5
Cost	960

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

Alternative 10
Error	26.0
Cost	960

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

Reproduce

herbie shell --seed 2022309 
(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))))