ENA, Section 1.4, Exercise 4a

Average Accuracy: 52.5% → 99.0%

Time: 19.0s

Precision: binary64

Cost: 6976

\[-1 \leq x \land x \leq 1\]

Math

\[\frac{x - \sin x}{\tan x} \]

↓

\[\frac{x}{\frac{\frac{\tan x}{x} \cdot 6}{x}} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

↓

(FPCore (x) :precision binary64 (/ x (/ (* (/ (tan x) x) 6.0) x)))

C

double code(double x) {
	return (x - sin(x)) / tan(x);
}

↓

double code(double x) {
	return x / (((tan(x) / x) * 6.0) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / tan(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (((tan(x) / x) * 6.0d0) / x)
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / Math.tan(x);
}

↓

public static double code(double x) {
	return x / (((Math.tan(x) / x) * 6.0) / x);
}

Python

def code(x):
	return (x - math.sin(x)) / math.tan(x)

↓

def code(x):
	return x / (((math.tan(x) / x) * 6.0) / x)

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

↓

function code(x)
	return Float64(x / Float64(Float64(Float64(tan(x) / x) * 6.0) / x))
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / tan(x);
end

↓

function tmp = code(x)
	tmp = x / (((tan(x) / x) * 6.0) / x);
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x / N[(N[(N[(N[Tan[x], $MachinePrecision] / x), $MachinePrecision] * 6.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Target

Original	52.5%
Target	98.8%
Herbie	99.0%

\[0.16666666666666666 \cdot \left(x \cdot x\right) \]

Derivation

1. Initial program 52.5%

   \[\frac{x - \sin x}{\tan x} \]

2. Taylor expanded in x around 0 83.2%

   \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot {x}^{3}}}{\tan x} \]

3. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{0.16666666666666666}{\frac{\tan x}{x}} \cdot \left(x \cdot x\right)} \]
   Proof

   [Start] 83.2	\[ \frac{0.16666666666666666 \cdot {x}^{3}}{\tan x} \]
   associate-/l* [=>] 83.2	\[ \color{blue}{\frac{0.16666666666666666}{\frac{\tan x}{{x}^{3}}}} \]
   cube-mult [=>] 83.2	\[ \frac{0.16666666666666666}{\frac{\tan x}{\color{blue}{x \cdot \left(x \cdot x\right)}}} \]
   associate-/r* [=>] 97.9	\[ \frac{0.16666666666666666}{\color{blue}{\frac{\frac{\tan x}{x}}{x \cdot x}}} \]
   associate-/r/ [=>] 98.9	\[ \color{blue}{\frac{0.16666666666666666}{\frac{\tan x}{x}} \cdot \left(x \cdot x\right)} \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{\frac{\tan x}{x} \cdot 6}{x}}} \]
   Proof

   [Start] 98.9	\[ \frac{0.16666666666666666}{\frac{\tan x}{x}} \cdot \left(x \cdot x\right) \]
   associate-*l/ [=>] 98.9	\[ \color{blue}{\frac{0.16666666666666666 \cdot \left(x \cdot x\right)}{\frac{\tan x}{x}}} \]
   *-commutative [=>] 98.9	\[ \frac{\color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666}}{\frac{\tan x}{x}} \]
   associate-/l* [=>] 99.0	\[ \color{blue}{\frac{x \cdot x}{\frac{\frac{\tan x}{x}}{0.16666666666666666}}} \]
   associate-/l* [=>] 99.0	\[ \color{blue}{\frac{x}{\frac{\frac{\frac{\tan x}{x}}{0.16666666666666666}}{x}}} \]
   div-inv [=>] 99.0	\[ \frac{x}{\frac{\color{blue}{\frac{\tan x}{x} \cdot \frac{1}{0.16666666666666666}}}{x}} \]
   metadata-eval [=>] 99.0	\[ \frac{x}{\frac{\frac{\tan x}{x} \cdot \color{blue}{6}}{x}} \]

5. Final simplification 99.0%

   \[\leadsto \frac{x}{\frac{\frac{\tan x}{x} \cdot 6}{x}} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	320

\[0.16666666666666666 \cdot \left(x \cdot x\right) \]

Alternative 2
Accuracy	98.9%
Cost	320

\[x \cdot \frac{x}{6} \]

Alternative 3
Accuracy	98.9%
Cost	320

\[\frac{x}{\frac{6}{x}} \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (* 0.16666666666666666 (* x x))

  (/ (- x (sin x)) (tan x)))