ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.3% → 100.0%

Time: 5.0s

Alternatives: 2

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: 53.3% 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 52.4%

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

   1. flip-- 52.4%

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

   2. div-inv 52.4%

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

   3. metadata-eval 52.4%

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

   4. 1-sub-cos 99.9%

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

   5. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. unpow2 99.9%

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

   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 2: 99.5% accurate, 20.6× 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 52.4%

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

3. Taylor expanded in x around 0 98.5%

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

   1. add-sqr-sqrt 98.1%

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

   2. pow2 98.1%

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

   3. *-commutative 98.1%

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

   4. sqrt-prod 98.0%

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

   5. sqrt-pow1 98.0%

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

   6. metadata-eval 98.0%

      \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{0.5}\right)}^{2} \]

   7. pow1 98.0%

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

5. Applied egg-rr 98.0%

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

   1. unpow-prod-down 97.6%

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

   2. unpow2 97.6%

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

   3. pow2 97.6%

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

   4. rem-square-sqrt 98.5%

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

   5. associate-*l* 98.5%

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

7. Applied egg-rr 98.5%

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

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 2024108 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :alt
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))