ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.1% → 99.6%

Time: 11.5s

Alternatives: 8

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: 99.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.6%

   \[\frac{x - \sin x}{\tan x} \]
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 99.7%

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{x - \sin x}{\tan x} \]
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 99.6%

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

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right) \cdot x, x, -0.06388888888888888\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \]
8. 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 2024230 
(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)))