ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.2% → 99.6%

Time: 15.9s

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: 53.2% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\\ x \cdot \frac{x \cdot \mathsf{fma}\left(t\_0 \cdot t\_0, t\_1 \cdot \left(t\_1 \cdot t\_1\right), 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, t\_1 \cdot \left(\left(x \cdot x\right) \cdot t\_1\right), 0.027777777777777776\right) - \left(x \cdot x\right) \cdot \left(t\_1 \cdot 0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1
         (fma
          (* x x)
          (fma (* x x) -0.00023644179894179894 -0.0007275132275132275)
          -0.06388888888888888)))
   (*
    x
    (/
     (* x (fma (* t_0 t_0) (* t_1 (* t_1 t_1)) 0.004629629629629629))
     (-
      (fma (* x x) (* t_1 (* (* x x) t_1)) 0.027777777777777776)
      (* (* x x) (* t_1 0.16666666666666666)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = fma((x * x), fma((x * x), -0.00023644179894179894, -0.0007275132275132275), -0.06388888888888888);
	return x * ((x * fma((t_0 * t_0), (t_1 * (t_1 * t_1)), 0.004629629629629629)) / (fma((x * x), (t_1 * ((x * x) * t_1)), 0.027777777777777776) - ((x * x) * (t_1 * 0.16666666666666666))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = fma(Float64(x * x), fma(Float64(x * x), -0.00023644179894179894, -0.0007275132275132275), -0.06388888888888888)
	return Float64(x * Float64(Float64(x * fma(Float64(t_0 * t_0), Float64(t_1 * Float64(t_1 * t_1)), 0.004629629629629629)) / Float64(fma(Float64(x * x), Float64(t_1 * Float64(Float64(x * x) * t_1)), 0.027777777777777776) - Float64(Float64(x * x) * Float64(t_1 * 0.16666666666666666)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894 + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]}, N[(x * N[(N[(x * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + 0.004629629629629629), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + 0.027777777777777776), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 54.1%

   \[\frac{x - \sin x}{\tan x} \]
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. 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 99.4%

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

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), 0.004629629629629629\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), 0.027777777777777776\right) - \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot 0.16666666666666666\right)}} \]

8. Final simplification 99.4%

   \[\leadsto x \cdot \frac{x \cdot \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), 0.004629629629629629\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), 0.027777777777777776\right) - \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot 0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\\ x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, t\_0, -0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          (* x x)
          (fma (* x x) -0.00023644179894179894 -0.0007275132275132275)
          -0.06388888888888888)))
   (*
    x
    (/
     (* x (fma (* x x) (* t_0 (* (* x x) t_0)) -0.027777777777777776))
     (fma (* x x) t_0 -0.16666666666666666)))))

C

double code(double x) {
	double t_0 = fma((x * x), fma((x * x), -0.00023644179894179894, -0.0007275132275132275), -0.06388888888888888);
	return x * ((x * fma((x * x), (t_0 * ((x * x) * t_0)), -0.027777777777777776)) / fma((x * x), t_0, -0.16666666666666666));
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), fma(Float64(x * x), -0.00023644179894179894, -0.0007275132275132275), -0.06388888888888888)
	return Float64(x * Float64(Float64(x * fma(Float64(x * x), Float64(t_0 * Float64(Float64(x * x) * t_0)), -0.027777777777777776)) / fma(Float64(x * x), t_0, -0.16666666666666666)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894 + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]}, N[(x * N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.1%

   \[\frac{x - \sin x}{\tan x} \]
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. 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 99.4%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)}} \]

8. Final simplification 99.4%

   \[\leadsto x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 3: 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 54.1%

   \[\frac{x - \sin x}{\tan x} \]
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. 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 99.4%

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

Alternative 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 54.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
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 99.3

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

5. Simplified 99.3%

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

Alternative 5: 99.3% accurate, 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 54.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
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 99.3

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

5. Simplified 99.3%

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

Alternative 6: 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 54.1%

   \[\frac{x - \sin x}{\tan x} \]
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 98.4

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

5. Simplified 98.4%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
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 98.4

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

   \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) \]
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 54.1%

   \[\frac{x - \sin x}{\tan x} \]
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 98.4

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

   \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]
7. 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 2024199 
(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)))