tanhf (example 3.4)

Percentage Accurate: 53.1% → 100.0%

Time: 9.2s

Alternatives: 10

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (tan (* x 0.5)))

C

double code(double x) {
	return tan((x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x * 0.5d0))
end function

Java

public static double code(double x) {
	return Math.tan((x * 0.5));
}

Python

def code(x):
	return math.tan((x * 0.5))

Julia

function code(x)
	return tan(Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = tan((x * 0.5));
end

Wolfram

code[x_] := N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{1 - \cos x}{\sin x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \cos x}}{\sin x} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\cos x}}{\sin x} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{1 - \cos x}{\color{blue}{\sin x}} \]

   5. hang-p0-tan N/A

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   6. lower-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   7. div-inv N/A

      \[\leadsto \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   8. metadata-eval N/A

      \[\leadsto \tan \left(x \cdot \color{blue}{\frac{1}{2}}\right) \]

   9. lower-*.f64 100.0

      \[\leadsto \tan \color{blue}{\left(x \cdot 0.5\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right)} \]
5. Add Preprocessing

Alternative 2: 52.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (*
    x
    (fma
     (fma x (* x 0.00042162698412698415) 0.004166666666666667)
     (* (* x x) (* x x))
     (fma x (* x 0.041666666666666664) 0.5)))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * fma(fma(x, (x * 0.00042162698412698415), 0.004166666666666667), ((x * x) * (x * x)), fma(x, (x * 0.041666666666666664), 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * fma(fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667), Float64(Float64(x * x) * Float64(x * x)), fma(x, Float64(x * 0.041666666666666664), 0.5)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{40320} \cdot {x}^{2} + \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{17}{40320} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{17}{40320} \cdot x\right) \cdot x} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{17}{40320} \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{17}{40320} \cdot x, \frac{1}{240}\right)}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{17}{40320}}, \frac{1}{240}\right), \frac{1}{24}\right), \frac{1}{2}\right) \]

      16. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.00042162698412698415}, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\right) \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 52.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), x \cdot \left(x \cdot x\right), x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (fma
    (fma
     x
     (* x (fma x (* x 0.00042162698412698415) 0.004166666666666667))
     0.041666666666666664)
    (* x (* x x))
    (* x 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = fma(fma(x, (x * fma(x, (x * 0.00042162698412698415), 0.004166666666666667)), 0.041666666666666664), (x * (x * x)), (x * 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = fma(fma(x, Float64(x * fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667)), 0.041666666666666664), Float64(x * Float64(x * x)), Float64(x * 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(N[(x * N[(x * N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision]), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{40320} \cdot {x}^{2} + \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{17}{40320} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{17}{40320} \cdot x\right) \cdot x} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{17}{40320} \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{17}{40320} \cdot x, \frac{1}{240}\right)}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{17}{40320}}, \frac{1}{240}\right), \frac{1}{24}\right), \frac{1}{2}\right) \]

      16. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.00042162698412698415}, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x \cdot 0.5\right) \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 52.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (*
    x
    (fma
     x
     (*
      x
      (fma
       x
       (* x (fma x (* x 0.00042162698412698415) 0.004166666666666667))
       0.041666666666666664))
     0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * fma(x, (x * fma(x, (x * fma(x, (x * 0.00042162698412698415), 0.004166666666666667)), 0.041666666666666664)), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * fma(x, Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667)), 0.041666666666666664)), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision]), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{40320} \cdot {x}^{2} + \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{17}{40320} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{17}{40320} \cdot x\right) \cdot x} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{17}{40320} \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{17}{40320} \cdot x, \frac{1}{240}\right)}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{17}{40320}}, \frac{1}{240}\right), \frac{1}{24}\right), \frac{1}{2}\right) \]

      16. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.00042162698412698415}, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right) \cdot \color{blue}{x} \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (*
    x
    (fma
     (* x x)
     (fma
      (* x x)
      (fma x (* x 0.00042162698412698415) 0.004166666666666667)
      0.041666666666666664)
     0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * fma((x * x), fma((x * x), fma(x, (x * 0.00042162698412698415), 0.004166666666666667), 0.041666666666666664), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667), 0.041666666666666664), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right)\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + {x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right), \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}\right) + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{240} + \frac{17}{40320} \cdot {x}^{2}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      10. +-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{40320} \cdot {x}^{2} + \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{17}{40320} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{17}{40320} \cdot x\right) \cdot x} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      13. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{17}{40320} \cdot x\right)} + \frac{1}{240}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{17}{40320} \cdot x, \frac{1}{240}\right)}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{17}{40320}}, \frac{1}{240}\right), \frac{1}{24}\right), \frac{1}{2}\right) \]

      16. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.00042162698412698415}, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)} \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), x \cdot \left(x \cdot x\right), x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (fma
    (fma x (* x 0.004166666666666667) 0.041666666666666664)
    (* x (* x x))
    (* x 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = fma(fma(x, (x * 0.004166666666666667), 0.041666666666666664), (x * (x * x)), (x * 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = fma(fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664), Float64(x * Float64(x * x)), Float64(x * 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{240} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{240} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{24}, \frac{1}{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{240} \cdot x\right) \cdot x} + \frac{1}{24}, \frac{1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{240} \cdot x\right)} + \frac{1}{24}, \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{240} \cdot x, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.004166666666666667}, 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x \cdot 0.5\right) \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 52.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2)
   (*
    x
    (fma (* x x) (fma x (* x 0.004166666666666667) 0.041666666666666664) 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * fma((x * x), fma(x, (x * 0.004166666666666667), 0.041666666666666664), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{240} \cdot {x}^{2}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right)} \]

      4. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{240} \cdot {x}^{2}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{240} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{240} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{24}, \frac{1}{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{240} \cdot x\right) \cdot x} + \frac{1}{24}, \frac{1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{240} \cdot x\right)} + \frac{1}{24}, \frac{1}{2}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{240} \cdot x, \frac{1}{24}\right)}, \frac{1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{240}}, \frac{1}{24}\right), \frac{1}{2}\right) \]

      12. lower-*.f64 66.6

          \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.004166666666666667}, 0.041666666666666664\right), 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)} \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2) (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * fma(x, (x * 0.041666666666666664), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * fma(x, Float64(x * 0.041666666666666664), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 38.3%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. unpow2 N/A

         \[\leadsto x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}\right) \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(\frac{1}{24} \cdot x\right)} + \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x, \frac{1}{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}}, \frac{1}{2}\right) \]

      8. lower-*.f64 66.6

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.041666666666666664}, 0.5\right) \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)} \]

   if 3.2000000000000002 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.4%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.4

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.6% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) (* x 0.5) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = x * 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = x * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(x * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = x * 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(x * 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 38.0%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 66.8

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 1.3999999999999999 < x

   1. Initial program 98.4%

      \[\frac{1 - \cos x}{\sin x} \]
   2. Add Preprocessing

   4. Applied rewrites 10.5%

      \[\leadsto \color{blue}{{1}^{-0.5}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

      2. pow-base-1 10.5

         \[\leadsto \color{blue}{1} \]

   6. Applied rewrites 10.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 7.0% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 54.3%

   \[\frac{1 - \cos x}{\sin x} \]
2. Add Preprocessing

4. Applied rewrites 6.6%

   \[\leadsto \color{blue}{{1}^{-0.5}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{1}^{\frac{-1}{2}}} \]

   2. pow-base-1 6.6

      \[\leadsto \color{blue}{1} \]

6. Applied rewrites 6.6%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (tan (/ x 2.0)))

C

double code(double x) {
	return tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0));
}

Python

def code(x):
	return math.tan((x / 2.0))

Julia

function code(x)
	return tan(Float64(x / 2.0))
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0));
end

Wolfram

code[x_] := N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :alt
  (! :herbie-platform default (tan (/ x 2)))

  (/ (- 1.0 (cos x)) (sin x)))