tanhf (example 3.4)

Percentage Accurate: 52.2% → 100.0%

Time: 9.0s

Alternatives: 4

Speedup: 22.8×

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: 52.2% 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 55.0%

   \[\frac{1 - \cos x}{\sin x} \]
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: 52.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 48.8%

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

7. Simplified 48.8%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. frac-2neg N/A

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

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

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

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) - \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{{\frac{1}{2}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}^{3}}}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) - \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{{\frac{1}{2}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}^{3}}}\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(0 - \color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) - \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{{\frac{1}{2}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}^{3}}}\right)\right) \]

   9. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right) - \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)\right)}{{\frac{1}{2}}^{3} + {\left(x \cdot \left(x \cdot \frac{1}{24}\right)\right)}^{3}}\right)}\right)\right) \]

9. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{\frac{0 - x}{0 - \frac{1}{0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \color{blue}{\left(\frac{1}{6} \cdot {x}^{2} - 2\right)}\right) \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{1}{6} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\frac{1}{6} \cdot {x}^{2}\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left({x}^{2} \cdot \frac{1}{6}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right), \left(\mathsf{neg}\left(2\right)\right)\right)\right) \]

    7. metadata-eval 49.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right), -2\right)\right) \]

12. Simplified 49.4%

    \[\leadsto \frac{0 - x}{\color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666 + -2}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{6}} + -2} \]

    2. distribute-frac-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{x}{\left(x \cdot x\right) \cdot \frac{1}{6} + -2}\right) \]

    3. distribute-frac-neg2 N/A

       \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(\left(\left(x \cdot x\right) \cdot \frac{1}{6} + -2\right)\right)}} \]

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

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

    5. +-commutative N/A

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

    6. distribute-neg-in N/A

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

    7. metadata-eval N/A

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

    8. unsub-neg N/A

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

    9. --lowering--.f64 N/A

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

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

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

    11. *-lowering-*.f64 49.4%

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

14. Applied egg-rr 49.4%

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

Alternative 3: 51.6% accurate, 22.8× speedup

Math

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

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(x * Float64(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[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 48.8%

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

7. Simplified 48.8%

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

Alternative 4: 51.7% 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 55.0%

   \[\frac{1 - \cos x}{\sin x} \]
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 48.7%

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

7. Simplified 48.7%

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

8. Final simplification 48.7%

   \[\leadsto x \cdot 0.5 \]
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 2024191 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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