ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.5% → 100.0%

Time: 5.1s

Alternatives: 4

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: 53.5% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin x) (tan (/ x 2.0))))

C

double code(double x) {
	return sin(x) * tan((x / 2.0));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sin(x) * Math.tan((x / 2.0));
}

Python

def code(x):
	return math.sin(x) * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(sin(x) * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sin(x) * tan((x / 2.0));
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

   1. flip-- 52.3%

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

   2. div-inv 52.3%

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

   3. metadata-eval 52.3%

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

   4. 1-sub-cos 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. associate-*r/ 100.0%

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

   4. hang-0p-tan 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

   \[\leadsto \sin x \cdot \tan \left(\frac{x}{2}\right) \]

Alternative 2: 100.0% accurate, 0.9× speedup

Math

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

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(x * Float64(x * fma(x, Float64(x * -0.041666666666666664), 0.5)))
end

Wolfram

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

Derivation

1. Initial program 52.3%

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

   1. flip-- 52.3%

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

   2. div-inv 52.3%

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

   3. metadata-eval 52.3%

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

   4. 1-sub-cos 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. associate-*r/ 100.0%

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

   4. hang-0p-tan 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. pow-sqr 99.9%

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

   4. associate-*r* 99.9%

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

   5. distribute-rgt-out 99.9%

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

   6. *-commutative 99.9%

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

   7. unpow2 99.9%

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

   8. unpow2 99.9%

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

   9. associate-*l* 99.9%

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

   10. fma-def 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around 0 99.9%

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

    1. +-commutative 99.9%

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

    2. metadata-eval 99.9%

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

    3. pow-sqr 99.9%

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

    4. unpow2 99.9%

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

    5. unpow2 99.9%

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

    6. unpow2 99.9%

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

    7. *-commutative 99.9%

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

    8. associate-*r* 99.9%

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

    9. associate-*r* 99.9%

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

    10. associate-*l* 99.9%

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

    11. *-commutative 99.9%

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

    12. associate-*l* 99.9%

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

    13. distribute-lft-out 99.9%

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

    14. associate-*r* 99.9%

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

    15. associate-*r* 99.9%

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

    16. cube-mult 99.9%

        \[\leadsto x \cdot \left(\color{blue}{{x}^{3}} \cdot -0.041666666666666664 + x \cdot 0.5\right) \]

    17. *-commutative 99.9%

        \[\leadsto x \cdot \left(\color{blue}{-0.041666666666666664 \cdot {x}^{3}} + x \cdot 0.5\right) \]

    18. fma-def 99.9%

        \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-0.041666666666666664, {x}^{3}, x \cdot 0.5\right)} \]

11. Simplified 99.9%

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(-0.041666666666666664, {x}^{3}, x \cdot 0.5\right)} \]

12. Taylor expanded in x around 0 99.9%

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

    1. unpow2 99.9%

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

    2. metadata-eval 99.9%

       \[\leadsto 0.5 \cdot \left(x \cdot x\right) + -0.041666666666666664 \cdot {x}^{\color{blue}{\left(3 + 1\right)}} \]

    3. pow-plus 99.9%

       \[\leadsto 0.5 \cdot \left(x \cdot x\right) + -0.041666666666666664 \cdot \color{blue}{\left({x}^{3} \cdot x\right)} \]

    4. unpow3 99.9%

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

    5. associate-*r* 99.9%

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

    6. associate-*r* 99.9%

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

    7. *-commutative 99.9%

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

    8. associate-*r* 99.9%

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

    9. distribute-rgt-out 99.9%

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

    10. +-commutative 99.9%

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

    11. fma-udef 99.9%

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

    12. associate-*r* 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.3%

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

   1. flip-- 52.3%

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

   2. div-inv 52.3%

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

   3. metadata-eval 52.3%

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

   4. 1-sub-cos 100.0%

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

3. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. associate-*r/ 100.0%

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

   4. hang-0p-tan 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.9%

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

   1. +-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. pow-sqr 99.9%

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

   4. associate-*r* 99.9%

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

   5. distribute-rgt-out 99.9%

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

   6. *-commutative 99.9%

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

   7. unpow2 99.9%

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

   8. unpow2 99.9%

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

   9. associate-*l* 99.9%

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

   10. fma-def 99.9%

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

8. Simplified 99.9%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]
9. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. distribute-rgt-in 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \left(x \cdot \left(x \cdot -0.041666666666666664\right)\right) \cdot \left(x \cdot x\right) + 0.5 \cdot \left(x \cdot x\right) \]

Alternative 4: 99.6% 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 52.3%

   \[1 - \cos x \]

2. Taylor expanded in x around 0 99.6%

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

   1. unpow2 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto 0.5 \cdot \left(x \cdot x\right) \]

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 2023178 
(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)))