ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.0% → 99.6%

Time: 14.2s

Alternatives: 10

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: 53.0% 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: 99.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   x
   0.16666666666666666
   (*
    (*
     (*
      (fma
       (fma (* x x) -0.00023644179894179894 -0.0007275132275132275)
       (* x x)
       -0.06388888888888888)
      x)
     x)
    x))
  x))

C

double code(double x) {
	return fma(x, 0.16666666666666666, (((fma(fma((x * x), -0.00023644179894179894, -0.0007275132275132275), (x * x), -0.06388888888888888) * x) * x) * x)) * x;
}

Julia

function code(x)
	return Float64(fma(x, 0.16666666666666666, Float64(Float64(Float64(fma(fma(Float64(x * x), -0.00023644179894179894, -0.0007275132275132275), Float64(x * x), -0.06388888888888888) * x) * x) * x)) * x)
end

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666 + N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894 + -0.0007275132275132275), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.06388888888888888), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (fma
     (fma -0.00023644179894179894 (* x x) -0.0007275132275132275)
     (* x x)
     -0.06388888888888888)
    (* x x)
    0.16666666666666666)
   x)
  x))

C

double code(double x) {
	return (fma(fma(fma(-0.00023644179894179894, (x * x), -0.0007275132275132275), (x * x), -0.06388888888888888), (x * x), 0.16666666666666666) * x) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(fma(-0.00023644179894179894, Float64(x * x), -0.0007275132275132275), Float64(x * x), -0.06388888888888888), Float64(x * x), 0.16666666666666666) * x) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(-0.00023644179894179894 * N[(x * x), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.06388888888888888), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0007275132275132275, x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (* (* (fma -0.0007275132275132275 (* x x) -0.06388888888888888) x) x)
   x
   (* 0.16666666666666666 x))
  x))

C

double code(double x) {
	return fma(((fma(-0.0007275132275132275, (x * x), -0.06388888888888888) * x) * x), x, (0.16666666666666666 * x)) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(fma(-0.0007275132275132275, Float64(x * x), -0.06388888888888888) * x) * x), x, Float64(0.16666666666666666 * x)) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(-0.0007275132275132275 * N[(x * x), $MachinePrecision] + -0.06388888888888888), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{6}, \left(\left(x \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \cdot x\right) \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(x \cdot x, -0.0007275132275132275, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0007275132275132275, x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(x \cdot x, -0.0007275132275132275, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   x
   0.16666666666666666
   (* (* (* (fma (* x x) -0.0007275132275132275 -0.06388888888888888) x) x) x))
  x))

C

double code(double x) {
	return fma(x, 0.16666666666666666, (((fma((x * x), -0.0007275132275132275, -0.06388888888888888) * x) * x) * x)) * x;
}

Julia

function code(x)
	return Float64(fma(x, 0.16666666666666666, Float64(Float64(Float64(fma(Float64(x * x), -0.0007275132275132275, -0.06388888888888888) * x) * x) * x)) * x)
end

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666 + N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.0007275132275132275 + -0.06388888888888888), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{6}, \left(\left(x \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \cdot x\right) \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(x \cdot x, -0.0007275132275132275, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0007275132275132275, x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (fma -0.0007275132275132275 (* x x) -0.06388888888888888)
    (* x x)
    0.16666666666666666)
   x)
  x))

C

double code(double x) {
	return (fma(fma(-0.0007275132275132275, (x * x), -0.06388888888888888), (x * x), 0.16666666666666666) * x) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(-0.0007275132275132275, Float64(x * x), -0.06388888888888888), Float64(x * x), 0.16666666666666666) * x) * x)
end

Wolfram

code[x_] := N[(N[(N[(N[(-0.0007275132275132275 * N[(x * x), $MachinePrecision] + -0.06388888888888888), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \cdot x} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \cdot x\right)} \cdot x \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \cdot x\right)} \cdot x \]

   7. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) + \frac{1}{6}\right)} \cdot x\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot {x}^{2}} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}, {x}^{2}, \frac{1}{6}\right)} \cdot x\right) \cdot x \]

   10. sub-neg N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{-11}{15120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{15120} \cdot {x}^{2} + \color{blue}{\frac{-23}{360}}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-11}{15120}, {x}^{2}, \frac{-23}{360}\right)}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-11}{15120}, \color{blue}{x \cdot x}, \frac{-23}{360}\right), {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-11}{15120}, \color{blue}{x \cdot x}, \frac{-23}{360}\right), {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   15. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-11}{15120}, x \cdot x, \frac{-23}{360}\right), \color{blue}{x \cdot x}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   16. lower-*.f64 99.3

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0007275132275132275, x \cdot x, -0.06388888888888888\right), \color{blue}{x \cdot x}, 0.16666666666666666\right) \cdot x\right) \cdot x \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0007275132275132275, x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Add Preprocessing

Alternative 6: 99.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 98.9

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

10. Applied rewrites 98.9%

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

12. Applied rewrites 98.9%

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

Alternative 7: 99.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(x, 0.16666666666666666, \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.9%

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

Alternative 8: 99.3% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\frac{x - \sin x}{\tan x} \]
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. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 9: 98.7% accurate, 19.5× speedup

Math

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

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(Float64(0.16666666666666666 * x) * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right) \cdot x} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{6} \cdot x\right) \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 98.1%

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

Alternative 10: 98.7% 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 52.4%

   \[\frac{x - \sin x}{\tan x} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 98.1

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

5. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} \]
6. Add Preprocessing

Developer Target 1: 98.7% 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 2024295 
(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)))