ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.1% → 99.6%

Time: 11.8s

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: 53.1% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right)\right)}^{-1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x x)
  (pow
   (fma
    (fma
     (fma -0.00023644179894179894 (* x x) -0.0007275132275132275)
     (* x x)
     -0.06388888888888888)
    (* x x)
    0.16666666666666666)
   -1.0)))

C

double code(double x) {
	return (x * x) / pow(fma(fma(fma(-0.00023644179894179894, (x * x), -0.0007275132275132275), (x * x), -0.06388888888888888), (x * x), 0.16666666666666666), -1.0);
}

Julia

function code(x)
	return Float64(Float64(x * x) / (fma(fma(fma(-0.00023644179894179894, Float64(x * x), -0.0007275132275132275), Float64(x * x), -0.06388888888888888), Float64(x * x), 0.16666666666666666) ^ -1.0))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] / N[Power[N[(N[(N[(-0.00023644179894179894 * N[(x * x), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.06388888888888888), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right)}}} \]

10. Final simplification 99.7%

    \[\leadsto \frac{x \cdot x}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right)\right)}^{-1}} \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 4.9× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \frac{x}{6}, \left(\left(-0.06388888888888888 \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x (/ x 6.0) (* (* (* -0.06388888888888888 x) x) (* x x))))

C

double code(double x) {
	return fma(x, (x / 6.0), (((-0.06388888888888888 * x) * x) * (x * x)));
}

Julia

function code(x)
	return fma(x, Float64(x / 6.0), Float64(Float64(Float64(-0.06388888888888888 * x) * x) * Float64(x * x)))
end

Wolfram

code[x_] := N[(x * N[(x / 6.0), $MachinePrecision] + N[(N[(N[(-0.06388888888888888 * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

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

12. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 9.8× speedup

Math

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

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(fma(-0.06388888888888888, Float64(x * x), 0.16666666666666666) * Float64(x * x))
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

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

Alternative 8: 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 55.5%

   \[\frac{x - \sin x}{\tan x} \]
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 9: 98.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{6} \end{array} \]

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(Float64(x * x) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) / 6.0;
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00023644179894179894, x \cdot x, -0.0007275132275132275\right), x \cdot x, -0.06388888888888888\right), x \cdot x, 0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.3%

    \[\leadsto \frac{x \cdot x}{6} \]
13. Add Preprocessing

Alternative 10: 98.8% 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 55.5%

   \[\frac{x - \sin x}{\tan x} \]
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 99.2

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

5. Applied rewrites 99.2%

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

Developer Target 1: 98.8% 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 2024305 
(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)))