ENA, Section 1.4, Mentioned, A

Percentage Accurate: 52.2% → 99.9%

Time: 4.1s

Alternatives: 3

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (0.5d0 + (x * (x * (-0.041666666666666664d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 + -0.041666666666666664 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 100.0%

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

   2. associate-*r* 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto {x}^{2} \cdot \left(0.5 + x \cdot \left(x \cdot -0.041666666666666664\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x 2.0) 0.5))

C

double code(double x) {
	return pow(x, 2.0) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * 0.5d0
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * 0.5;
}

Python

def code(x):
	return math.pow(x, 2.0) * 0.5

Julia

function code(x)
	return Float64((x ^ 2.0) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * 0.5;
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 53.3%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]

4. Final simplification 99.8%

   \[\leadsto {x}^{2} \cdot 0.5 \]
5. Add Preprocessing

Alternative 3: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

   \[1 - \cos x \]
2. Add Preprocessing
3. Add Preprocessing

Developer target: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sin x}{1 + \cos x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (sin x) (sin x)) (+ 1.0 (cos x))))

C

double code(double x) {
	return (sin(x) * sin(x)) / (1.0 + cos(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) * sin(x)) / (1.0d0 + cos(x))
end function

Java

public static double code(double x) {
	return (Math.sin(x) * Math.sin(x)) / (1.0 + Math.cos(x));
}

Python

def code(x):
	return (math.sin(x) * math.sin(x)) / (1.0 + math.cos(x))

Julia

function code(x)
	return Float64(Float64(sin(x) * sin(x)) / Float64(1.0 + cos(x)))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) * sin(x)) / (1.0 + cos(x));
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :alt
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))