tanhf (example 3.4)

Percentage Accurate: 52.8% → 100.0%

Time: 10.4s

Alternatives: 10

Speedup: 6.3×

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: 52.8% 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 52.3%

   \[\frac{1 - \cos x}{\sin x} \]
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: 52.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right)\\ \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(x \cdot t\_0, x \cdot \left(x \cdot 0.004166666666666667\right), -0.001736111111111111\right)}{\mathsf{fma}\left(x, t\_0, -0.041666666666666664\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x (* x 0.00042162698412698415) 0.004166666666666667))))
   (if (<= x 2.45)
     (*
      x
      (fma
       x
       (/
        (*
         x
         (fma
          (* x t_0)
          (* x (* x 0.004166666666666667))
          -0.001736111111111111))
        (fma x t_0 -0.041666666666666664))
       0.5))
     1.0)))

C

double code(double x) {
	double t_0 = x * fma(x, (x * 0.00042162698412698415), 0.004166666666666667);
	double tmp;
	if (x <= 2.45) {
		tmp = x * fma(x, ((x * fma((x * t_0), (x * (x * 0.004166666666666667)), -0.001736111111111111)) / fma(x, t_0, -0.041666666666666664)), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667))
	tmp = 0.0
	if (x <= 2.45)
		tmp = Float64(x * fma(x, Float64(Float64(x * fma(Float64(x * t_0), Float64(x * Float64(x * 0.004166666666666667)), -0.001736111111111111)) / fma(x, t_0, -0.041666666666666664)), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.45], N[(x * N[(x * N[(N[(x * N[(N[(x * t$95$0), $MachinePrecision] * N[(x * N[(x * 0.004166666666666667), $MachinePrecision]), $MachinePrecision] + -0.001736111111111111), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 36.4%

      \[\frac{1 - \cos x}{\sin x} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 68.0%

      \[\leadsto x \cdot \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right)\right), -0.001736111111111111\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), -0.041666666666666664\right)}}, 0.5\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto x \cdot \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{17}{40320}, \frac{1}{240}\right)\right), x \cdot \left(x \cdot \frac{1}{240}\right), \frac{-1}{576}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{17}{40320}, \frac{1}{240}\right), \frac{-1}{24}\right)}, \frac{1}{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.2%

       \[\leadsto x \cdot \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right)\right), x \cdot \left(x \cdot 0.004166666666666667\right), -0.001736111111111111\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), -0.041666666666666664\right)}, 0.5\right) \]

   if 2.4500000000000002 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.7

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

   6. Applied rewrites 10.7%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right)\right), x \cdot \left(x \cdot 0.004166666666666667\right), -0.001736111111111111\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), -0.041666666666666664\right)}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), x \cdot x, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (fma
    (*
     x
     (fma
      (* x x)
      (fma x (* x 0.00042162698412698415) 0.004166666666666667)
      0.041666666666666664))
    (* x x)
    (* x 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = fma((x * fma((x * x), fma(x, (x * 0.00042162698412698415), 0.004166666666666667), 0.041666666666666664)), (x * x), (x * 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.00042162698412698415), 0.004166666666666667), 0.041666666666666664)), Float64(x * x), Float64(x * 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.00042162698412698415), $MachinePrecision] + 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 68.3%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 4: 52.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.00042162698412698415, 0.004166666666666667\right), 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (*
    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 <= 3.1) {
		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 <= 3.1)
		tmp = Float64(x * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.00042162698412698415, 0.004166666666666667), 0.041666666666666664)), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.00042162698412698415 + 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 68.3%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 5: 52.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (fma
    (* x (* x x))
    (fma x (* x 0.004166666666666667) 0.041666666666666664)
    (* x 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = fma((x * (x * x)), fma(x, (x * 0.004166666666666667), 0.041666666666666664), (x * 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(Float64(x * Float64(x * x)), fma(x, Float64(x * 0.004166666666666667), 0.041666666666666664), Float64(x * 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   7. Applied rewrites 68.2%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 6: 52.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.004166666666666667, 0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (*
    x
    (fma x (* x (fma x (* x 0.004166666666666667) 0.041666666666666664)) 0.5))
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		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 <= 3.1)
		tmp = Float64(x * fma(x, Float64(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, 3.1], N[(x * N[(x * N[(x * N[(x * N[(x * 0.004166666666666667), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 7: 52.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 0.041666666666666664, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) (fma (* x (* x x)) 0.041666666666666664 (* x 0.5)) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = fma((x * (x * x)), 0.041666666666666664, (x * 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = fma(Float64(x * Float64(x * x)), 0.041666666666666664, Float64(x * 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 68.1%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 8: 52.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, x \cdot 0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) (* x (fma x (* x 0.041666666666666664) 0.5)) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x * fma(x, (x * 0.041666666666666664), 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		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, 3.1], 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 < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

Alternative 9: 52.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.1) (* x 0.5) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		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 <= 3.1d0) 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 <= 3.1) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = x * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = Float64(x * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = x * 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(x * 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   if 3.10000000000000009 < x

   1. Initial program 98.9%

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

   4. Applied rewrites 10.6%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 10.6

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

   6. Applied rewrites 10.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 6.8% 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 52.3%

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

4. Applied rewrites 6.9%

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

   1. lift-pow.f64 N/A

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

   2. pow-base-1 6.9

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

6. Applied rewrites 6.9%

   \[\leadsto \color{blue}{1} \]
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 2024235 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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