Average Error: 29.7 → 0.0
Time: 5.2s
Precision: binary64
\[-0.01 \leq x \land x \leq 0.01\]
\[1 - \cos x \]
↓
\[\sin x \cdot \tan \left(\frac{x}{2}\right) \]
(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)

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.0
\[\frac{\sin x \cdot \sin x}{1 + \cos x} \]

Derivation

  1. Initial program 29.7

    \[1 - \cos x \]

  2. Applied flip--_binary6429.7

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}} \]

  3. Simplified0.0

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

  4. Taylor expanded in x around inf 0.0

    \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{1 + \cos x}} \]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(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)))