ENA, Section 1.4, Exercise 4a

Percentage Accurate: 52.8% → 99.5%

Time: 11.5s

Alternatives: 7

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: 52.8% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\\ \frac{\mathsf{fma}\left({t\_0}^{3}, {x}^{6}, 0.004629629629629629\right) \cdot x}{\mathsf{fma}\left({t\_0}^{2}, {x}^{4}, 0.027777777777777776\right) - \left(t\_0 \cdot x\right) \cdot \left(0.16666666666666666 \cdot x\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* -0.0007275132275132275 (* x x)) 0.06388888888888888)))
   (*
    (/
     (* (fma (pow t_0 3.0) (pow x 6.0) 0.004629629629629629) x)
     (-
      (fma (pow t_0 2.0) (pow x 4.0) 0.027777777777777776)
      (* (* t_0 x) (* 0.16666666666666666 x))))
    x)))

C

double code(double x) {
	double t_0 = (-0.0007275132275132275 * (x * x)) - 0.06388888888888888;
	return ((fma(pow(t_0, 3.0), pow(x, 6.0), 0.004629629629629629) * x) / (fma(pow(t_0, 2.0), pow(x, 4.0), 0.027777777777777776) - ((t_0 * x) * (0.16666666666666666 * x)))) * x;
}

Julia

function code(x)
	t_0 = Float64(Float64(-0.0007275132275132275 * Float64(x * x)) - 0.06388888888888888)
	return Float64(Float64(Float64(fma((t_0 ^ 3.0), (x ^ 6.0), 0.004629629629629629) * x) / Float64(fma((t_0 ^ 2.0), (x ^ 4.0), 0.027777777777777776) - Float64(Float64(t_0 * x) * Float64(0.16666666666666666 * x)))) * x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(-0.0007275132275132275 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.06388888888888888), $MachinePrecision]}, N[(N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision] + 0.004629629629629629), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(t$95$0 * x), $MachinePrecision] * N[(0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Initial program 54.9%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left({\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right)}^{3}, {x}^{6}, 0.004629629629629629\right) \cdot x}{\mathsf{fma}\left({\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right)}^{2}, {x}^{4}, 0.027777777777777776\right) - \left(\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot x\right)} \cdot x \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot -0.0007275132275132275 - 0.06388888888888888\right) \cdot x\right) \cdot x\\ \frac{\left({t\_0}^{2} - 0.027777777777777776\right) \cdot x}{t\_0 - 0.16666666666666666} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* (- (* (* x x) -0.0007275132275132275) 0.06388888888888888) x)
          x)))
   (*
    (/
     (* (- (pow t_0 2.0) 0.027777777777777776) x)
     (- t_0 0.16666666666666666))
    x)))

C

double code(double x) {
	double t_0 = ((((x * x) * -0.0007275132275132275) - 0.06388888888888888) * x) * x;
	return (((pow(t_0, 2.0) - 0.027777777777777776) * x) / (t_0 - 0.16666666666666666)) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = ((((x * x) * (-0.0007275132275132275d0)) - 0.06388888888888888d0) * x) * x
    code = ((((t_0 ** 2.0d0) - 0.027777777777777776d0) * x) / (t_0 - 0.16666666666666666d0)) * x
end function

Java

public static double code(double x) {
	double t_0 = ((((x * x) * -0.0007275132275132275) - 0.06388888888888888) * x) * x;
	return (((Math.pow(t_0, 2.0) - 0.027777777777777776) * x) / (t_0 - 0.16666666666666666)) * x;
}

Python

def code(x):
	t_0 = ((((x * x) * -0.0007275132275132275) - 0.06388888888888888) * x) * x
	return (((math.pow(t_0, 2.0) - 0.027777777777777776) * x) / (t_0 - 0.16666666666666666)) * x

Julia

function code(x)
	t_0 = Float64(Float64(Float64(Float64(Float64(x * x) * -0.0007275132275132275) - 0.06388888888888888) * x) * x)
	return Float64(Float64(Float64(Float64((t_0 ^ 2.0) - 0.027777777777777776) * x) / Float64(t_0 - 0.16666666666666666)) * x)
end

MATLAB

function tmp = code(x)
	t_0 = ((((x * x) * -0.0007275132275132275) - 0.06388888888888888) * x) * x;
	tmp = ((((t_0 ^ 2.0) - 0.027777777777777776) * x) / (t_0 - 0.16666666666666666)) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.0007275132275132275), $MachinePrecision] - 0.06388888888888888), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 0.027777777777777776), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Initial program 54.9%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left({\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right)}^{3}, {x}^{6}, 0.004629629629629629\right) \cdot x}{\mathsf{fma}\left({\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right)}^{2}, {x}^{4}, 0.027777777777777776\right) - \left(\left(-0.0007275132275132275 \cdot \left(x \cdot x\right) - 0.06388888888888888\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot x\right)} \cdot x \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\left({\left(\left(\left(\left(x \cdot x\right) \cdot -0.0007275132275132275 - 0.06388888888888888\right) \cdot x\right) \cdot x\right)}^{2} - 0.027777777777777776\right) \cdot x}{\left(\left(\left(x \cdot x\right) \cdot -0.0007275132275132275 - 0.06388888888888888\right) \cdot x\right) \cdot x - 0.16666666666666666} \cdot x \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\left(-0.00023644179894179894 \cdot \left(x \cdot x\right) - 0.0007275132275132275\right) \cdot x\right) \cdot x - 0.06388888888888888, \left(x \cdot x\right) \cdot 0.16666666666666666\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.9%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 5: 99.4% 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 54.9%

   \[\frac{x - \sin x}{\tan x} \]
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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 98.8% 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 54.9%

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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

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 99.1%

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

Alternative 7: 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 54.9%

   \[\frac{x - \sin x}{\tan x} \]
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 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} \]
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 2024338 
(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)))