ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.1% → 99.4%

Time: 12.4s

Alternatives: 6

Speedup: 19.5×

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: 53.1% 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: 99.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ x (/ 1.0 (fma (* x x) -0.06388888888888888 0.16666666666666666)))))

C

double code(double x) {
	return x * (x / (1.0 / fma((x * x), -0.06388888888888888, 0.16666666666666666)));
}

Julia

function code(x)
	return Float64(x * Float64(x / Float64(1.0 / fma(Float64(x * x), -0.06388888888888888, 0.16666666666666666))))
end

Wolfram

code[x_] := N[(x * N[(x / N[(1.0 / N[(N[(x * x), $MachinePrecision] * -0.06388888888888888 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-23}{360} + \frac{1}{6}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}}\right) \]

   3. clear-num N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}}\right) \]

   4. un-div-inv N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}} \]

   6. clear-num N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}}}} \]

   7. flip-+ N/A

      \[\leadsto x \cdot \frac{x}{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{-23}{360} + \frac{1}{6}}}} \]

   8. lift-fma.f64 N/A

      \[\leadsto x \cdot \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-23}{360}, \frac{1}{6}\right)}}} \]

   9. lower-/.f64 99.8

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}}} \]

7. Applied rewrites 99.8%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}}} \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 3: 98.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x 6.0)))

C

double code(double x) {
	return x * (x / 6.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / 6.0d0)
end function

Java

public static double code(double x) {
	return x * (x / 6.0);
}

Python

def code(x):
	return x * (x / 6.0)

Julia

function code(x)
	return Float64(x * Float64(x / 6.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / 6.0);
end

Wolfram

code[x_] := N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-23}{360} + \frac{1}{6}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}}\right) \]

   3. clear-num N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}}\right) \]

   4. un-div-inv N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}}} \]

   6. clear-num N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}}}} \]

   7. flip-+ N/A

      \[\leadsto x \cdot \frac{x}{\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{-23}{360} + \frac{1}{6}}}} \]

   8. lift-fma.f64 N/A

      \[\leadsto x \cdot \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-23}{360}, \frac{1}{6}\right)}}} \]

   9. lower-/.f64 99.8

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}}} \]

7. Applied rewrites 99.8%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}}} \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \frac{x}{\color{blue}{6}} \]
9. Step-by-step derivation

10. Applied rewrites 99.6%

    \[\leadsto x \cdot \frac{x}{\color{blue}{6}} \]
11. Add Preprocessing

Alternative 4: 98.6% accurate, 19.5× 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 52.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} \]

   3. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 98.6% accurate, 19.5× speedup

Math

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

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(Float64(x * x) * 0.16666666666666666)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 4.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0.43478260869565216 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.43478260869565216)

C

double code(double x) {
	return 0.43478260869565216;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.43478260869565216d0
end function

Java

public static double code(double x) {
	return 0.43478260869565216;
}

Python

def code(x):
	return 0.43478260869565216

Julia

function code(x)
	return 0.43478260869565216
end

MATLAB

function tmp = code(x)
	tmp = 0.43478260869565216;
end

Wolfram

code[x_] := 0.43478260869565216

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-23}{360} + \frac{1}{6}\right)\right) \]

   2. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \frac{-23}{360}, \frac{1}{6}\right) \cdot x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-23}{360}, \frac{1}{6}\right)\right) \cdot x} \]

   5. lift-fma.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360} + \frac{1}{6}\right)}\right) \cdot x \]

   6. flip-+ N/A

      \[\leadsto \left(x \cdot \color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}}\right) \cdot x \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}\right)}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}} \cdot x \]

   8. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}\right)\right) \cdot x}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x\right)} \cdot x}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\left(\left(\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x\right) \cdot x}{\left(x \cdot x\right) \cdot \frac{-23}{360} - \frac{1}{6}}} \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\left(x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.00408179012345679, -0.027777777777777776\right)\right) \cdot x}{\mathsf{fma}\left(x, x \cdot -0.06388888888888888, -0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\left(\frac{-1}{36} \cdot x\right)} \cdot x}{\mathsf{fma}\left(x, x \cdot \frac{-23}{360}, \frac{-1}{6}\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{-1}{36}\right)} \cdot x}{\mathsf{fma}\left(x, x \cdot \frac{-23}{360}, \frac{-1}{6}\right)} \]

   2. lower-*.f64 99.7

      \[\leadsto \frac{\color{blue}{\left(x \cdot -0.027777777777777776\right)} \cdot x}{\mathsf{fma}\left(x, x \cdot -0.06388888888888888, -0.16666666666666666\right)} \]

10. Applied rewrites 99.7%

    \[\leadsto \frac{\color{blue}{\left(x \cdot -0.027777777777777776\right)} \cdot x}{\mathsf{fma}\left(x, x \cdot -0.06388888888888888, -0.16666666666666666\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{10}{23}} \]
12. Step-by-step derivation

13. Applied rewrites 4.2%

    \[\leadsto \color{blue}{0.43478260869565216} \]
14. Add Preprocessing

Developer Target 1: 98.6% accurate, 19.5× 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 2024214 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (! :herbie-platform default (* 1/6 (* x x)))

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