tanhf (example 3.4)

Percentage Accurate: 53.0% → 100.0%

Time: 13.5s

Alternatives: 5

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: 53.0% 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

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

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]

Derivation

1. Initial program 52.4%

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

   1. hang-p0-tan N/A

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

   2. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 51.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{2 + \left(x \cdot x\right) \cdot -0.16666666666666666}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (+ 2.0 (* (* x x) -0.16666666666666666)) x)))

C

double code(double x) {
	return 1.0 / ((2.0 + ((x * x) * -0.16666666666666666)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((2.0d0 + ((x * x) * (-0.16666666666666666d0))) / x)
end function

Java

public static double code(double x) {
	return 1.0 / ((2.0 + ((x * x) * -0.16666666666666666)) / x);
}

Python

def code(x):
	return 1.0 / ((2.0 + ((x * x) * -0.16666666666666666)) / x)

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(2.0 + Float64(Float64(x * x) * -0.16666666666666666)) / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((2.0 + ((x * x) * -0.16666666666666666)) / x);
end

Wolfram

code[x_] := N[(1.0 / N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

   1. hang-p0-tan N/A

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

   2. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 51.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 51.9%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot x\right) \cdot x\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   11. *-lowering-*.f64 51.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

9. Applied egg-rr 51.9%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) + 0.5 \cdot x} \]
10. Step-by-step derivation

    1. flip3-+ N/A

       \[\leadsto \frac{{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}}{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}} \]

    2. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}{{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}}}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}{{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}}\right)}\right) \]

    4. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}^{3} + {\left(\frac{1}{2} \cdot x\right)}^{3}}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) + \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\frac{1}{2} \cdot x\right) - \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right) \cdot \left(\frac{1}{2} \cdot x\right)\right)}}}\right)\right) \]

    5. flip3-+ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) + \color{blue}{\frac{1}{2} \cdot x}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) + \frac{1}{2} \cdot x\right)}\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) + x \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \]

    8. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{24}\right) + \frac{1}{2}\right)}\right)\right)\right) \]

11. Applied egg-rr 51.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right) + 0.5\right)}}} \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{2 + \frac{-1}{6} \cdot {x}^{2}}{x}\right)}\right) \]
13. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 + \frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{x}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right), x\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), x\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), x\right)\right) \]

    6. *-lowering-*.f64 52.4%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right), x\right)\right) \]

14. Simplified 52.4%

    \[\leadsto \frac{1}{\color{blue}{\frac{2 + \left(x \cdot x\right) \cdot -0.16666666666666666}{x}}} \]
15. Add Preprocessing

Alternative 3: 50.7% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) + x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* x (* x (* x 0.041666666666666664))) (* x 0.5)))

C

double code(double x) {
	return (x * (x * (x * 0.041666666666666664))) + (x * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * (x * 0.041666666666666664d0))) + (x * 0.5d0)
end function

Java

public static double code(double x) {
	return (x * (x * (x * 0.041666666666666664))) + (x * 0.5);
}

Python

def code(x):
	return (x * (x * (x * 0.041666666666666664))) + (x * 0.5)

Julia

function code(x)
	return Float64(Float64(x * Float64(x * Float64(x * 0.041666666666666664))) + Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * (x * 0.041666666666666664))) + (x * 0.5);
end

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. hang-p0-tan N/A

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

   2. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 51.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 51.9%

   \[\leadsto \color{blue}{x \cdot \left(0.5 + 0.041666666666666664 \cdot \left(x \cdot x\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot \left(x \cdot x\right)\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot x\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\frac{1}{24} \cdot x\right) \cdot x\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{24} \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \left(\frac{1}{2} \cdot x\right)\right) \]

   11. *-lowering-*.f64 51.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{24}\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right)\right) \]

9. Applied egg-rr 51.9%

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

10. Final simplification 51.9%

    \[\leadsto x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right) + x \cdot 0.5 \]
11. Add Preprocessing

Alternative 4: 50.7% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 0.5 (* (* x x) 0.041666666666666664))))

C

double code(double x) {
	return x * (0.5 + ((x * x) * 0.041666666666666664));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.5d0 + ((x * x) * 0.041666666666666664d0))
end function

Java

public static double code(double x) {
	return x * (0.5 + ((x * x) * 0.041666666666666664));
}

Python

def code(x):
	return x * (0.5 + ((x * x) * 0.041666666666666664))

Julia

function code(x)
	return Float64(x * Float64(0.5 + Float64(Float64(x * x) * 0.041666666666666664)))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.5 + ((x * x) * 0.041666666666666664));
end

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. hang-p0-tan N/A

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

   2. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

   5. *-lowering-*.f64 51.9%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

7. Simplified 51.9%

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

8. Final simplification 51.9%

   \[\leadsto x \cdot \left(0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right) \]
9. Add Preprocessing

Alternative 5: 50.9% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

   1. hang-p0-tan N/A

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

   2. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 51.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{x}\right) \]

7. Simplified 51.7%

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

8. Final simplification 51.7%

   \[\leadsto x \cdot 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.0× 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 2024141 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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