ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.1% → 99.9%

Time: 4.1s

Alternatives: 3

Speedup: 20.6×

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: 53.1% 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, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x * (x * (0.5 + ((x * x) * -0.041666666666666664)));
}

Python

def code(x):
	return x * (x * (0.5 + ((x * x) * -0.041666666666666664)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{24} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{24}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{24}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{24}\right)\right)\right)\right) \]

5. Simplified 100.0%

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

Alternative 2: 99.5% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.5 * (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 * (x * x);
}

Python

def code(x):
	return 0.5 * (x * x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
6. Add Preprocessing

Alternative 3: 52.1% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 53.4%

   \[\leadsto 1 - \color{blue}{1} \]
6. Step-by-step derivation

   1. metadata-eval 53.4%

      \[\leadsto 0 \]

7. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 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 2024158 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :alt
  (! :herbie-platform default (/ (* (sin x) (sin x)) (+ 1 (cos x))))

  (- 1.0 (cos x)))