ENA, Section 1.4, Exercise 4a

Percentage Accurate: 54.4% → 99.3%

Time: 14.3s

Alternatives: 3

Speedup: 41.0×

Specification

\[-1 \leq x \land x \leq 1\]

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{\tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{\tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (tan x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / tan(x))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot -0.06388888888888888\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (+ 0.16666666666666666 (* x (* x -0.06388888888888888))))))

C

double code(double x) {
	return x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + (x * (x * (-0.06388888888888888d0)))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(x * Float64(x * -0.06388888888888888)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.16666666666666666 + N[(x * N[(x * -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 51.8%

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

   1. +-commutative 51.8%

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

   2. mul-1-neg 51.8%

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

   3. unsub-neg 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in x around 0 99.6%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(0.16666666666666666 + -0.06388888888888888 \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 99.6%

      \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + -0.06388888888888888 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   2. associate-*r* 99.6%

      \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + \color{blue}{\left(-0.06388888888888888 \cdot x\right) \cdot x}\right)\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + \color{blue}{\left(-0.06388888888888888 \cdot x\right) \cdot x}\right)\right) \]

9. Final simplification 99.6%

   \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot -0.06388888888888888\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 98.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.16666666666666666)))

C

double code(double x) {
	return x * (x * 0.16666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 0.16666666666666666d0)
end function

Java

public static double code(double x) {
	return x * (x * 0.16666666666666666);
}

Python

def code(x):
	return x * (x * 0.16666666666666666)

Julia

function code(x)
	return Float64(x * Float64(x * 0.16666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.16666666666666666);
end

Wolfram

code[x_] := N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 51.8%

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

   1. +-commutative 51.8%

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

   2. mul-1-neg 51.8%

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

   3. unsub-neg 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in x around 0 99.0%

   \[\leadsto x \cdot \color{blue}{\left(0.16666666666666666 \cdot x\right)} \]

7. Final simplification 99.0%

   \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) \]
8. Add Preprocessing

Alternative 3: 4.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 52.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 4.2%

   \[\leadsto \frac{\color{blue}{x}}{\tan x} \]

4. Taylor expanded in x around 0 4.2%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Developer target: 98.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.16666666666666666 (* x x)))

C

double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.16666666666666666d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Python

def code(x):
	return 0.16666666666666666 * (x * x)

Julia

function code(x)
	return Float64(0.16666666666666666 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.16666666666666666 * (x * x);
end

Wolfram

code[x_] := N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (* 0.16666666666666666 (* x x))

  (/ (- x (sin x)) (tan x)))