Average Error: 30.2 → 0.0

Time: 6.9s

Precision: binary64

Cost: 13120

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

\[1 - \cos x \]

↓

\[\sin x \cdot \tan \left(\frac{x}{2}\right) \]

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

↓

(FPCore (x) :precision binary64 (* (sin x) (tan (/ x 2.0))))

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

↓

double code(double x) {
	return sin(x) * tan((x / 2.0));
}

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 = sin(x) * tan((x / 2.0d0))
end function

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

↓

public static double code(double x) {
	return Math.sin(x) * Math.tan((x / 2.0));
}

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

↓

def code(x):
	return math.sin(x) * math.tan((x / 2.0))

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

↓

function code(x)
	return Float64(sin(x) * tan(Float64(x / 2.0)))
end

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

↓

function tmp = code(x)
	tmp = sin(x) * tan((x / 2.0));
end

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

↓

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

1 - \cos x

↓

\sin x \cdot \tan \left(\frac{x}{2}\right)

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	30.2
Target	0.0
Herbie	0.0

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

Derivation

Initial program 30.2
\[1 - \cos x \]
Applied egg-rr0.0
\[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{1 + \cos x}} \]
Taylor expanded in x around inf 0.0
\[\leadsto \color{blue}{\frac{{\sin x}^{2}}{1 + \cos x}} \]
Simplified0.0
\[\leadsto \color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)} \]
Proof
(*.f64 (sin.f64 x) (tan.f64 (/.f64 x 2))): 0 points increase in error, 0 points decrease in error
(*.f64 (sin.f64 x) (Rewrite<= hang-0p-tan_binary64 (/.f64 (sin.f64 x) (+.f64 1 (cos.f64 x))))): 1 points increase in error, 1 points decrease in error
(Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sin.f64 x) (sin.f64 x)) (+.f64 1 (cos.f64 x)))): 2 points increase in error, 0 points decrease in error
(/.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 x) 2)) (+.f64 1 (cos.f64 x))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \sin x \cdot \tan \left(\frac{x}{2}\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	6976

\[x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\right) \]

Alternative 2
Error	0.0
Cost	832

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

Alternative 3
Error	0.3
Cost	320

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

Reproduce

herbie shell --seed 2022291 
(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)))