ENA, Section 1.4, Mentioned, A

Average Accuracy: 52.0% → 99.4%

Time: 5.3s

Precision: binary64

Cost: 320

\[-0.01 \leq x \land x \leq 0.01\]

Math

\[1 - \cos x \]

↓

\[x \cdot \left(x \cdot 0.5\right) \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cos x)))

↓

(FPCore (x) :precision binary64 (* x (* x 0.5)))

C

double code(double x) {
	return 1.0 - cos(x);
}

↓

double code(double x) {
	return x * (x * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 0.5d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.cos(x);
}

↓

public static double code(double x) {
	return x * (x * 0.5);
}

Python

def code(x):
	return 1.0 - math.cos(x)

↓

def code(x):
	return x * (x * 0.5)

Julia

function code(x)
	return Float64(1.0 - cos(x))
end

↓

function code(x)
	return Float64(x * Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - cos(x);
end

↓

function tmp = code(x)
	tmp = x * (x * 0.5);
end

Wolfram

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Target

Original	52.0%
Target	100.0%
Herbie	99.4%

\[\frac{\sin x \cdot \sin x}{1 + \cos x} \]

Derivation

1. Initial program 52.0%

   \[1 - \cos x \]

2. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
   Proof

   [Start] 99.4	\[ 0.5 \cdot {x}^{2} \]
   unpow2 [=>] 99.4	\[ 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

4. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot x\right)} \]
   Proof

   [Start] 99.4	\[ 0.5 \cdot {x}^{2} \]
   unpow2 [=>] 99.4	\[ 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]
   associate-*r* [=>] 99.4	\[ \color{blue}{\left(0.5 \cdot x\right) \cdot x} \]
   *-commutative [=>] 99.4	\[ \color{blue}{x \cdot \left(0.5 \cdot x\right)} \]

6. Final simplification 99.4%

   \[\leadsto x \cdot \left(x \cdot 0.5\right) \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	320

\[0.5 \cdot \left(x \cdot x\right) \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :herbie-target
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))