Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 0.4

Time: 38.0s

Precision: binary64

Cost: 20288

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{\sin k}\\ 2 \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{\cos k}{t}\right)\right) \end{array} \]

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 (/ (/ l k) (sin k)))) (* 2.0 (* t_1 (* t_1 (/ (cos k) t))))))

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 = (l / k) / sin(k);
	return 2.0 * (t_1 * (t_1 * (cos(k) / t)));
}

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
    t_1 = (l / k) / sin(k)
    code = 2.0d0 * (t_1 * (t_1 * (cos(k) / t)))
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 = (l / k) / Math.sin(k);
	return 2.0 * (t_1 * (t_1 * (Math.cos(k) / t)));
}

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 = (l / k) / math.sin(k)
	return 2.0 * (t_1 * (t_1 * (math.cos(k) / t)))

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(l / k) / sin(k))
	return Float64(2.0 * Float64(t_1 * Float64(t_1 * Float64(cos(k) / t))))
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)
	t_1 = (l / k) / sin(k);
	tmp = 2.0 * (t_1 * (t_1 * (cos(k) / t)));
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[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(t$95$1 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / t), $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. Simplified 40.6

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
   Proof
   (*.f64 (/.f64 (/.f64 (/.f64 2 (tan.f64 k)) (pow.f64 t 3)) (sin.f64 k)) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 2 (tan.f64 k)) (*.f64 (pow.f64 t 3) (sin.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 2 points decrease in error
   (*.f64 (Rewrite=> associate-/l/_binary64 (/.f64 2 (*.f64 (*.f64 (pow.f64 t 3) (sin.f64 k)) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 4 points decrease in error
   (*.f64 (/.f64 2 (Rewrite=> associate-*l*_binary64 (*.f64 (pow.f64 t 3) (*.f64 (sin.f64 k) (tan.f64 k))))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 1 points increase in error, 2 points decrease in error
   (*.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k)))) (/.f64 (*.f64 l l) (pow.f64 (/.f64 k t) 2))): 3 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 (/.f64 k t) 2) 0)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (+.f64 (pow.f64 (/.f64 k t) 2) (Rewrite<= metadata-eval (-.f64 1 1))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (pow.f64 (/.f64 k t) 2) 1) 1)))): 41 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 (sin.f64 k) (tan.f64 k))) (/.f64 (*.f64 l l) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (pow.f64 (/.f64 k t) 2))) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 2 (pow.f64 t 3)) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 5 points increase in error, 2 points decrease in error
   (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 (pow.f64 t 3) (*.f64 l l)))) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))): 1 points increase in error, 3 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 1 points increase in error, 1 points decrease in error
   (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (-.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 0 points increase in error, 2 points decrease in error
   (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 2 points increase in error, 0 points decrease in error

3. Taylor expanded in k around inf 22.6

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

4. Applied egg-rr 23.7

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

5. Applied egg-rr 8.2

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	5.5
Cost	20816

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t}{\cos k}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t_2}\\ \mathbf{elif}\;\ell \leq -1.2240903767286312 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_1}{\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t}\right)}}\\ \mathbf{elif}\;\ell \leq 9.375709162923945 \cdot 10^{-226}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;\ell \leq 10^{+105}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{k \cdot \frac{t_2}{\frac{\ell}{k}}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{\sin k \cdot \frac{k}{\frac{\ell}{\sin k}}}}{t_2}\\ \end{array} \]

Alternative 2
Error	5.1
Cost	20752

\[\begin{array}{l} t_1 := \frac{t}{\cos k}\\ t_2 := \frac{2}{\frac{{\sin k}^{2}}{\frac{\ell}{k \cdot \frac{t_1}{\frac{\ell}{k}}}}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t_1}\\ \mathbf{elif}\;\ell \leq -1.4849181367107017 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 9.375709162923945 \cdot 10^{-226}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{{\left(\frac{k}{\frac{\ell}{\sin k}}\right)}^{2}}}{t_1}\\ \mathbf{elif}\;\ell \leq 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \end{array} \]

Alternative 3
Error	5.5
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{t}{\cos k}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+140}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t_3}\\ \mathbf{elif}\;\ell \leq -1.2240903767286312 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_2}{\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t}\right)}}\\ \mathbf{elif}\;\ell \leq 9.375709162923945 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 10^{+110}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{k \cdot \frac{t_3}{\frac{\ell}{k}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	6.3
Cost	20488

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{-108}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;k \leq 1.3709227987954457 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 5
Error	5.8
Cost	20232

\[\begin{array}{l} t_1 := 2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	5.8
Cost	20232

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;k \leq 10^{-12}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{{\left(\frac{k \cdot \sin k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 7
Error	5.8
Cost	20232

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-120}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2}}{t}\right)\\ \mathbf{elif}\;k \leq 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{\frac{t}{\cos k}}\\ \end{array} \]

Alternative 8
Error	10.7
Cost	14408

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

Alternative 9
Error	22.2
Cost	13956

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-313}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell} \cdot \frac{k}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 10
Error	22.6
Cost	13704

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)}\\ \mathbf{elif}\;t \leq 3.067796529322022 \cdot 10^{+79}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k \cdot \sin k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot \cos k} \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 11
Error	23.3
Cost	7616

\[\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \frac{1}{\frac{\ell}{k} \cdot \frac{\cos k}{t}}\right)} \]

Alternative 12
Error	23.4
Cost	1088

\[\frac{1}{\frac{k}{\frac{\ell}{k}} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\frac{2}{t}}\right)} \]

Alternative 13
Error	26.1
Cost	960

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

Alternative 14
Error	24.8
Cost	960

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

Alternative 15
Error	24.0
Cost	960

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

Alternative 16
Error	33.8
Cost	832

\[2 \cdot \frac{\ell \cdot \ell}{\frac{k \cdot t}{\frac{-0.16666666666666666}{k}}} \]

Reproduce

herbie shell --seed 2022297 
(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))))