ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.4% → 100.0%

Time: 6.9s

Alternatives: 6

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: 53.4% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\cos \left(\frac{x}{-2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (sin (* 0.5 x)) (sin x)) (cos (/ x -2.0))))

C

double code(double x) {
	return (sin((0.5 * x)) * sin(x)) / cos((x / -2.0));
}

Fortran

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

Java

public static double code(double x) {
	return (Math.sin((0.5 * x)) * Math.sin(x)) / Math.cos((x / -2.0));
}

Python

def code(x):
	return (math.sin((0.5 * x)) * math.sin(x)) / math.cos((x / -2.0))

Julia

function code(x)
	return Float64(Float64(sin(Float64(0.5 * x)) * sin(x)) / cos(Float64(x / -2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (sin((0.5 * x)) * sin(x)) / cos((x / -2.0));
end

Wolfram

code[x_] := N[(N[(N[Sin[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(x / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. metadata-eval 50.1%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. unpow2 100.0%

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

   4. associate-/l* 100.0%

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

   5. hang-0p-tan 100.0%

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

6. Simplified 100.0%

   \[\leadsto \color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)} \]
7. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. tan-quot 100.0%

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

   3. associate-*l/ 100.0%

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

   4. clear-num 99.9%

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

   5. associate-/r/ 100.0%

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

   6. metadata-eval 100.0%

      \[\leadsto \frac{\sin \left(\color{blue}{0.5} \cdot x\right) \cdot \sin x}{\cos \left(\frac{x}{2}\right)} \]

   7. frac-2neg 100.0%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\cos \color{blue}{\left(\frac{-x}{-2}\right)}} \]

   8. distribute-frac-neg 100.0%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\cos \color{blue}{\left(-\frac{x}{-2}\right)}} \]

   9. cos-neg 100.0%

      \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\color{blue}{\cos \left(\frac{x}{-2}\right)}} \]

   10. metadata-eval 100.0%

       \[\leadsto \frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\cos \left(\frac{x}{\color{blue}{-2}}\right)} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right) \cdot \sin x}{\cos \left(\frac{x}{-2}\right)}} \]
9. Add Preprocessing

Alternative 2: 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 50.1%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. metadata-eval 50.1%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. unpow2 100.0%

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

   4. associate-/l* 100.0%

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

   5. hang-0p-tan 100.0%

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

6. Simplified 100.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (0.5 * x) * math.sin(x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. metadata-eval 50.1%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. unpow2 100.0%

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

   4. associate-/l* 100.0%

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

   5. hang-0p-tan 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 20.6× speedup

Math

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

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(x * Float64(0.5 * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. metadata-eval 50.1%

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

   4. 1-sub-cos 100.0%

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

   5. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. unpow2 100.0%

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

   4. associate-/l* 100.0%

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

   5. hang-0p-tan 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 99.8%

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

8. Taylor expanded in x around 0 99.8%

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

Alternative 5: 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 50.1%

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

3. Taylor expanded in x around 0 99.8%

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

   1. unpow2 99.8%

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

5. Applied egg-rr 99.8%

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

Alternative 6: 52.5% 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 50.1%

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

3. Taylor expanded in x around 0 49.8%

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

   1. metadata-eval 49.8%

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

5. Applied egg-rr 49.8%

   \[\leadsto \color{blue}{0} \]
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 2024188 
(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)))