Toniolo and Linder, Equation (10-)

Average Error: 47.5 → 0.8

Time: 26.6s

Precision: binary64

Cost: 13760

Math

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

↓

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

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 (tan k)) (/ k l)) (/ (/ (/ l k) t) (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 / tan(k)) / (k / l)) * (((l / k) / t) / 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 / tan(k)) / (k / l)) * (((l / k) / t) / 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.tan(k)) / (k / l)) * (((l / k) / t) / 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.tan(k)) / (k / l)) * (((l / k) / t) / 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(Float64(Float64(2.0 / tan(k)) / Float64(k / l)) * Float64(Float64(Float64(l / k) / t) / 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 / tan(k)) / (k / l)) * (((l / k) / t) / 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[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.5

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

2. Simplified 40.1

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{{t}^{3}}}{\sin k} \cdot \frac{\ell \cdot \ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
   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))): 5 points increase in error, 6 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))): 6 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))): 2 points increase in error, 9 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, 3 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)))): 30 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)))): 6 points increase in error, 8 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))): 2 points increase in error, 5 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))))): 6 points increase in error, 4 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)))): 3 points increase in error, 3 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))): 3 points increase in error, 2 points decrease in error

3. Applied egg-rr 33.1

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

4. Taylor expanded in t around 0 22.0

   \[\leadsto \frac{1}{\frac{\sin k}{\frac{2}{\tan k} \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}}} \]

5. Simplified 6.1

   \[\leadsto \frac{1}{\frac{\sin k}{\frac{2}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)}}} \]
   Proof
   (*.f64 (/.f64 l k) (/.f64 l (*.f64 k t))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k (*.f64 k t)))): 72 points increase in error, 25 points decrease in error
   (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k (*.f64 k t))): 0 points increase in error, 0 points decrease in error
   (/.f64 (pow.f64 l 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 k k) t))): 20 points increase in error, 7 points decrease in error
   (/.f64 (pow.f64 l 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) t)): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.9

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k} \cdot \frac{\ell}{k}}{1} \cdot \frac{\frac{\frac{\ell}{k}}{t}}{\sin k}} \]

7. Applied egg-rr 0.8

   \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{k}{\ell}}}}{1} \cdot \frac{\frac{\frac{\ell}{k}}{t}}{\sin k} \]

8. Final simplification 0.8

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

Alternatives

Alternative 1
Error	13.9
Cost	14024

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

Alternative 2
Error	13.9
Cost	14024

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

Alternative 3
Error	13.8
Cost	14024

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

Alternative 4
Error	9.9
Cost	14024

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

Alternative 5
Error	10.9
Cost	14024

\[\begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot \frac{\frac{\tan k \cdot \sin k}{\frac{\ell}{t}}}{\ell}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	22.7
Cost	13960

\[\begin{array}{l} t_1 := \frac{\frac{2}{k} \cdot \frac{\ell}{t}}{k \cdot \frac{{\sin k}^{2}}{\ell}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.8
Cost	13760

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

Alternative 8
Error	22.4
Cost	7620

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-218}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t}}{\sin k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(\frac{2}{k \cdot k} + 0.6666666666666666\right)\\ \end{array} \]

Alternative 9
Error	23.8
Cost	7432

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{t}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.4
Cost	7432

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \cdot \left(\frac{2}{k \cdot k} + -0.3333333333333333\right)\\ \mathbf{if}\;k \leq -1 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-120}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	24.6
Cost	1348

\[\begin{array}{l} t_1 := \frac{2}{k \cdot k}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-143}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(t_1 + -0.3333333333333333\right)\\ \end{array} \]

Alternative 12
Error	24.5
Cost	1348

\[\begin{array}{l} t_1 := \frac{2}{k \cdot k}\\ \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-225}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\frac{t}{\frac{\ell}{k \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right) \cdot \left(t_1 + 0.6666666666666666\right)\\ \end{array} \]

Alternative 13
Error	26.1
Cost	960

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

Alternative 14
Error	25.8
Cost	960

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

Alternative 15
Error	25.6
Cost	960

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

Alternative 16
Error	32.8
Cost	704

\[\ell \cdot \left(\frac{\ell}{t} \cdot \frac{-0.3333333333333333}{k \cdot k}\right) \]

Reproduce

herbie shell --seed 2022325 
(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))))