ENA, Section 1.4, Mentioned, A

Average Error: 30.5 → 0.0

Time: 6.0s

Precision: binary64

Cost: 20096

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

Math

\[1 - \cos x \]

↓

\[0.5 \cdot {x}^{2} + \left(-0.041666666666666664 \cdot {x}^{4} + 0.001388888888888889 \cdot {x}^{6}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.5 (pow x 2.0))
  (+
   (* -0.041666666666666664 (pow x 4.0))
   (* 0.001388888888888889 (pow x 6.0)))))

C

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

↓

double code(double x) {
	return (0.5 * pow(x, 2.0)) + ((-0.041666666666666664 * pow(x, 4.0)) + (0.001388888888888889 * pow(x, 6.0)));
}

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 = (0.5d0 * (x ** 2.0d0)) + (((-0.041666666666666664d0) * (x ** 4.0d0)) + (0.001388888888888889d0 * (x ** 6.0d0)))
end function

Java

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

↓

public static double code(double x) {
	return (0.5 * Math.pow(x, 2.0)) + ((-0.041666666666666664 * Math.pow(x, 4.0)) + (0.001388888888888889 * Math.pow(x, 6.0)));
}

Python

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

↓

def code(x):
	return (0.5 * math.pow(x, 2.0)) + ((-0.041666666666666664 * math.pow(x, 4.0)) + (0.001388888888888889 * math.pow(x, 6.0)))

Julia

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

↓

function code(x)
	return Float64(Float64(0.5 * (x ^ 2.0)) + Float64(Float64(-0.041666666666666664 * (x ^ 4.0)) + Float64(0.001388888888888889 * (x ^ 6.0))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (0.5 * (x ^ 2.0)) + ((-0.041666666666666664 * (x ^ 4.0)) + (0.001388888888888889 * (x ^ 6.0)));
end

Wolfram

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

↓

code[x_] := N[(N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.041666666666666664 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	30.5
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 30.5

   \[1 - \cos x \]

2. Taylor expanded in x around 0 0.0

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2} + \left(-0.041666666666666664 \cdot {x}^{4} + 0.001388888888888889 \cdot {x}^{6}\right)} \]

3. Final simplification 0.0

   \[\leadsto 0.5 \cdot {x}^{2} + \left(-0.041666666666666664 \cdot {x}^{4} + 0.001388888888888889 \cdot {x}^{6}\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.3
Cost	6656

\[0.5 \cdot {x}^{2} \]

Alternative 3
Error	30.5
Cost	6592

\[1 - \cos x \]

Reproduce

herbie shell --seed 2023104 
(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)))