tanhf (example 3.4)

Percentage Accurate: 52.6% → 100.0%

Time: 6.0s

Alternatives: 3

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: 52.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

3. Taylor expanded in x around inf

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

   1. hang-p0-tan N/A

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

   2. *-rgt-identity N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 51.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00042162698412698415, x \cdot x, 0.004166666666666667\right), x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.00042162698412698415 (* x x) 0.004166666666666667)
    (* x x)
    0.041666666666666664)
   (* x x)
   0.5)
  x))

C

double code(double x) {
	return fma(fma(fma(0.00042162698412698415, (x * x), 0.004166666666666667), (x * x), 0.041666666666666664), (x * x), 0.5) * x;
}

Julia

function code(x)
	return Float64(fma(fma(fma(0.00042162698412698415, Float64(x * x), 0.004166666666666667), Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5) * x)
end

Wolfram

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

Derivation

1. Initial program 54.0%

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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 50.3

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

5. Applied rewrites 50.3%

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

Alternative 3: 51.3% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

   \[\frac{1 - \cos x}{\sin x} \]
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 50.3

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

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. 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 2024340 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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