Average Error: 29.7 → 0.0
Time: 4.7s
Precision: binary64
Cost: 960
\[-0.01 \leq x \land x \leq 0.01\]
\[1 - \cos x \]
↓
\[\left(x \cdot \left(x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot 0.5 \]
(FPCore (x) :precision binary64 (- 1.0 (cos x)))
↓
(FPCore (x)
 :precision binary64
 (+ (* (* x (* x -0.041666666666666664)) (* x x)) (* (* x x) 0.5)))
double code(double x) {
	return 1.0 - cos(x);
}
↓
double code(double x) {
	return ((x * (x * -0.041666666666666664)) * (x * x)) + ((x * x) * 0.5);
}

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

public static double code(double x) {
	return 1.0 - Math.cos(x);
}
↓
public static double code(double x) {
	return ((x * (x * -0.041666666666666664)) * (x * x)) + ((x * x) * 0.5);
}

def code(x):
	return 1.0 - math.cos(x)
↓
def code(x):
	return ((x * (x * -0.041666666666666664)) * (x * x)) + ((x * x) * 0.5)

function code(x)
	return Float64(1.0 - cos(x))
end
↓
function code(x)
	return Float64(Float64(Float64(x * Float64(x * -0.041666666666666664)) * Float64(x * x)) + Float64(Float64(x * x) * 0.5))
end

function tmp = code(x)
	tmp = 1.0 - cos(x);
end
↓
function tmp = code(x)
	tmp = ((x * (x * -0.041666666666666664)) * (x * x)) + ((x * x) * 0.5);
end

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[(N[(x * N[(x * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

1 - \cos x
↓
\left(x \cdot \left(x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot 0.5

Error

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 egg-rr0.0

    \[\leadsto \color{blue}{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}} \]

  3. 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, 0 points decrease in error
    (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sin.f64 x) (sin.f64 x)) (+.f64 1 (cos.f64 x)))): 3 points increase in error, 1 points decrease in error
    (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 x)) 1)) (+.f64 1 (cos.f64 x))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 (sin.f64 x) (sin.f64 x)) (/.f64 1 (+.f64 1 (cos.f64 x))))): 1 points increase in error, 1 points decrease in error

  4. Taylor expanded in x around 0 0.0

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

  5. Simplified0.0

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]
    Proof
    (*.f64 (*.f64 x x) (fma.f64 x (*.f64 x -1/24) 1/2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (fma.f64 x (*.f64 x -1/24) 1/2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 x 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 x -1/24)) 1/2))): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 x 2) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) -1/24)) 1/2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 x 2) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) -1/24) 1/2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 x 2) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/24 (pow.f64 x 2))) 1/2)): 0 points increase in error, 0 points decrease in error
    (*.f64 (pow.f64 x 2) (Rewrite=> +-commutative_binary64 (+.f64 1/2 (*.f64 -1/24 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 1/2 (pow.f64 x 2)) (*.f64 (*.f64 -1/24 (pow.f64 x 2)) (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/2 (pow.f64 x 2)) (Rewrite<= associate-*r*_binary64 (*.f64 -1/24 (*.f64 (pow.f64 x 2) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/2 (pow.f64 x 2)) (*.f64 -1/24 (Rewrite=> pow-sqr_binary64 (pow.f64 x (*.f64 2 2))))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 1/2 (pow.f64 x 2)) (*.f64 -1/24 (pow.f64 x (Rewrite=> metadata-eval 4)))): 0 points increase in error, 0 points decrease in error

  6. Applied egg-rr0.0

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

  7. Final simplification0.0

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

Alternatives

Alternative 1
Error0.3
Cost320
\[\left(x \cdot x\right) \cdot 0.5 \]
Alternative 2
Error0.3
Cost320
\[x \cdot \left(x \cdot 0.5\right) \]

Error

Reproduce

herbie shell --seed 2022338 
(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)))