ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.8% → 99.6%

Time: 13.0s

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: 53.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.7%

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

   1. distribute-rgt-in N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 99.7

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

7. Applied egg-rr 99.7%

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

Alternative 2: 99.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

5. Simplified 99.7%

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

Alternative 3: 99.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 99.7

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

5. Simplified 99.7%

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

   1. associate-*r* N/A

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

   2. distribute-rgt-in N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 99.7

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

7. Applied egg-rr 99.7%

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

Alternative 4: 99.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 99.7

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

5. Simplified 99.7%

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

Alternative 5: 99.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{\mathsf{fma}\left(x, x \cdot 2.3, 6\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x x) (fma x (* x 2.3) 6.0)))

C

double code(double x) {
	return (x * x) / fma(x, (x * 2.3), 6.0);
}

Julia

function code(x)
	return Float64(Float64(x * x) / fma(x, Float64(x * 2.3), 6.0))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] / N[(x * N[(x * 2.3), $MachinePrecision] + 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 99.6

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

5. Simplified 99.6%

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

   1. associate-*r* N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. clear-num N/A

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

   8. flip-+ N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate-*l* N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 99.6

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot x}{\color{blue}{6 + \frac{23}{10} \cdot {x}^{2}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{23}{10} \cdot {x}^{2} + 6}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x \cdot x}{\frac{23}{10} \cdot \color{blue}{\left(x \cdot x\right)} + 6} \]

   3. associate-*r* N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\left(\frac{23}{10} \cdot x\right) \cdot x} + 6} \]

   4. *-commutative N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{x \cdot \left(\frac{23}{10} \cdot x\right)} + 6} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{fma}\left(x, \frac{23}{10} \cdot x, 6\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{23}{10}}, 6\right)} \]

   7. *-lowering-*.f64 99.6

      \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(x, \color{blue}{x \cdot 2.3}, 6\right)} \]

10. Simplified 99.6%

    \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2.3, 6\right)}} \]
11. Add Preprocessing

Alternative 6: 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 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 99.6

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

5. Simplified 99.6%

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

Alternative 7: 98.8% 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 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 99.6

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

5. Simplified 99.6%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

   7. /-lowering-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 99.7

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0

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

10. Simplified 99.3%

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

Alternative 8: 98.7% 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 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.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. *-lowering-*.f64 99.1

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

5. Simplified 99.1%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.2

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 9: 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 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.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. *-lowering-*.f64 99.1

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 10: 4.2% 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 51.7%

   \[\frac{x - \sin x}{\tan x} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 99.6

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

5. Simplified 99.6%

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

   1. associate-*r* N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. clear-num N/A

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

   8. flip-+ N/A

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

   9. /-lowering-/.f64 N/A

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

   10. associate-*l* N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 99.6

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot x}{\color{blue}{6 + \frac{23}{10} \cdot {x}^{2}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{23}{10} \cdot {x}^{2} + 6}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x \cdot x}{\frac{23}{10} \cdot \color{blue}{\left(x \cdot x\right)} + 6} \]

   3. associate-*r* N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\left(\frac{23}{10} \cdot x\right) \cdot x} + 6} \]

   4. *-commutative N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{x \cdot \left(\frac{23}{10} \cdot x\right)} + 6} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{fma}\left(x, \frac{23}{10} \cdot x, 6\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{23}{10}}, 6\right)} \]

   7. *-lowering-*.f64 99.6

      \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(x, \color{blue}{x \cdot 2.3}, 6\right)} \]

10. Simplified 99.6%

    \[\leadsto \frac{x \cdot x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 2.3, 6\right)}} \]

11. Taylor expanded in x around inf

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

13. Simplified 4.3%

    \[\leadsto \color{blue}{0.43478260869565216} \]
14. 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 2024204 
(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)))