ENA, Section 1.4, Mentioned, A

Percentage Accurate: 52.5% → 99.9%

Time: 8.3s

Alternatives: 4

Speedup: 9.5×

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: 52.5% 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, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{24}, \frac{1}{2}\right) \]

   8. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

   \[\leadsto x \cdot \left(x \cdot 0.5\right) \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[1 - \cos x \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

   \[\leadsto \left(x \cdot x\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 51.6% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.6%

   \[\leadsto 1 - \color{blue}{1} \]
6. 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 2024223 
(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)))