ENA, Section 1.4, Mentioned, A

Average Error: 30.1 → 0.1

Time: 15.6s

Precision: binary64

Cost: 832

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

Math

\[1 - \cos x \]

↓

\[x \cdot \left(\left(-0.041666666666666664 \cdot \left(x \cdot x\right)\right) \cdot x + 0.5 \cdot x\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* x (+ (* (* -0.041666666666666664 (* x x)) x) (* 0.5 x))))

C

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

↓

double code(double x) {
	return x * (((-0.041666666666666664 * (x * x)) * x) + (0.5 * x));
}

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 * ((((-0.041666666666666664d0) * (x * x)) * x) + (0.5d0 * x))
end function

Java

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

↓

public static double code(double x) {
	return x * (((-0.041666666666666664 * (x * x)) * x) + (0.5 * x));
}

Python

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

↓

def code(x):
	return x * (((-0.041666666666666664 * (x * x)) * x) + (0.5 * x))

Julia

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

↓

function code(x)
	return Float64(x * Float64(Float64(Float64(-0.041666666666666664 * Float64(x * x)) * x) + Float64(0.5 * x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = x * (((-0.041666666666666664 * (x * x)) * x) + (0.5 * x));
end

Wolfram

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

↓

code[x_] := N[(x * N[(N[(N[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	30.1
Target	0.0
Herbie	0.1

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

Derivation

1. Initial program 30.1

   \[1 - \cos x \]

2. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2} + -0.041666666666666664 \cdot {x}^{4}} \]

3. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -0.041666666666666664 \cdot {x}^{4}\right)} \]
   Proof

4. Applied egg-rr 0.1

   \[\leadsto \color{blue}{x \cdot \left(\left(-0.041666666666666664 \cdot \left(x \cdot x\right)\right) \cdot x + 0.5 \cdot x\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	320

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

Alternative 2
Error	0.3
Cost	320

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

Alternative 3
Error	30.7
Cost	192

\[x - x \]

Reproduce

herbie shell --seed 2023010 
(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)))