ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.1% → 99.9%

Time: 4.8s

Alternatives: 5

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot {x}^{4}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x 0.5) x (* -0.041666666666666664 (pow x 4.0))))

C

double code(double x) {
	return fma((x * 0.5), x, (-0.041666666666666664 * pow(x, 4.0)));
}

Julia

function code(x)
	return fma(Float64(x * 0.5), x, Float64(-0.041666666666666664 * (x ^ 4.0)))
end

Wolfram

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

Derivation

1. Initial program 54.6%

   \[1 - \cos x \]
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%

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

5. Applied egg-rr 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. pow2 100.0%

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

   3. associate-*r* 100.0%

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

   4. fma-define 100.0%

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

   5. *-commutative 100.0%

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

   6. pow2 100.0%

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

   7. associate-*l* 100.0%

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

   8. pow-prod-up 100.0%

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

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot {x}^{\color{blue}{4}}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.5, x, -0.041666666666666664 \cdot {x}^{4}\right)} \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

   \[1 - \cos x \]
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%

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

5. Applied egg-rr 100.0%

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

   1. pow2 100.0%

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

7. Applied egg-rr 100.0%

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

Alternative 3: 99.5% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 54.6%

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}} \]

   2. clear-num 54.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + \cos x}{1 \cdot 1 - \cos x \cdot \cos x}}} \]

   3. metadata-eval 54.6%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{1} - \cos x \cdot \cos x}} \]

   4. 1-sub-cos 98.8%

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

   5. pow2 98.8%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{{\sin x}^{2}}}} \]

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \cos x}{{\sin x}^{2}}}} \]

5. Taylor expanded in x around 0 98.3%

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

   1. associate-/r/ 99.4%

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

   2. metadata-eval 99.4%

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

   3. pow2 99.4%

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

   4. associate-*r* 99.5%

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

   5. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

Alternative 4: 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 54.6%

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

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. unpow2 100.0%

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

5. Applied egg-rr 99.4%

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

Alternative 5: 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 54.6%

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

3. Taylor expanded in x around 0 53.4%

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

   1. metadata-eval 53.4%

      \[\leadsto \color{blue}{0} \]

5. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{0} \]
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 2024116 -o generate:simplify
(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)))