tanhf (example 3.4)

Percentage Accurate: 52.7% → 100.0%

Time: 7.2s

Alternatives: 7

Speedup: 17.9×

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: 52.7% 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 47.6%

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

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

   8. associate-/r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 53.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10.2:\\ \;\;\;\;\left(\frac{x \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.002857142857142857, x \cdot x, -2.4\right), x \cdot x, 24\right)} + 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 10.2)
   (*
    (+
     (/ (* x x) (fma (fma -0.002857142857142857 (* x x) -2.4) (* x x) 24.0))
     0.5)
    x)
   1.0))

C

double code(double x) {
	double tmp;
	if (x <= 10.2) {
		tmp = (((x * x) / fma(fma(-0.002857142857142857, (x * x), -2.4), (x * x), 24.0)) + 0.5) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 10.2)
		tmp = Float64(Float64(Float64(Float64(x * x) / fma(fma(-0.002857142857142857, Float64(x * x), -2.4), Float64(x * x), 24.0)) + 0.5) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 10.2], N[(N[(N[(N[(x * x), $MachinePrecision] / N[(N[(-0.002857142857142857 * N[(x * x), $MachinePrecision] + -2.4), $MachinePrecision] * N[(x * x), $MachinePrecision] + 24.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 10.199999999999999

   1. Initial program 32.0%

      \[\frac{1 - \cos x}{\sin x} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.00042162698412698415, x \cdot x, 0.004166666666666667\right), x \cdot x, 0.041666666666666664\right)}}, x \cdot x, 0.5\right) \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{24 + {x}^{2} \cdot \left(\frac{-1}{350} \cdot {x}^{2} - \frac{12}{5}\right)}, x \cdot x, \frac{1}{2}\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 72.2%

       \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.002857142857142857, x \cdot x, -2.4\right), x \cdot x, 24\right)}, x \cdot x, 0.5\right) \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 72.2%

       \[\leadsto \left(\frac{x \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(-0.002857142857142857, x \cdot x, -2.4\right), x \cdot x, 24\right)} + 0.5\right) \cdot x \]

   if 10.199999999999999 < x

   1. Initial program 98.6%

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

   4. Applied rewrites 11.5%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 11.5

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

   6. Applied rewrites 11.5%

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

Alternative 3: 53.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14:\\ \;\;\;\;\left(\frac{x \cdot x}{\mathsf{fma}\left(x \cdot x, -2.4, 24\right)} + 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 14.0) (* (+ (/ (* x x) (fma (* x x) -2.4 24.0)) 0.5) x) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 14.0) {
		tmp = (((x * x) / fma((x * x), -2.4, 24.0)) + 0.5) * x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 14.0)
		tmp = Float64(Float64(Float64(Float64(x * x) / fma(Float64(x * x), -2.4, 24.0)) + 0.5) * x);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 14.0], N[(N[(N[(N[(x * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -2.4 + 24.0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 14

   1. Initial program 32.0%

      \[\frac{1 - \cos x}{\sin x} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.00042162698412698415, x \cdot x, 0.004166666666666667\right), x \cdot x, 0.041666666666666664\right)}}, x \cdot x, 0.5\right) \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{24 + \frac{-12}{5} \cdot {x}^{2}}, x \cdot x, \frac{1}{2}\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 72.1%

       \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(-2.4, x \cdot x, 24\right)}, x \cdot x, 0.5\right) \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 72.1%

       \[\leadsto \left(\frac{x \cdot x}{\mathsf{fma}\left(x \cdot x, -2.4, 24\right)} + 0.5\right) \cdot x \]

   if 14 < x

   1. Initial program 98.6%

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

   4. Applied rewrites 11.5%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 11.5

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

   6. Applied rewrites 11.5%

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

Alternative 4: 53.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 3.2000000000000002 < x

   1. Initial program 98.6%

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

   4. Applied rewrites 11.5%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 11.5

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

   6. Applied rewrites 11.5%

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

Alternative 5: 52.9% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 3.2000000000000002 < x

   1. Initial program 98.6%

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

   4. Applied rewrites 11.5%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 11.5

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

   6. Applied rewrites 11.5%

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

Alternative 6: 52.9% accurate, 17.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 0.5 * x;
	} 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 = 0.5d0 * x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 31.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   if 1.3999999999999999 < x

   1. Initial program 98.6%

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

   4. Applied rewrites 11.7%

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

      1. lift-pow.f64 N/A

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

      2. pow-base-1 11.7

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

   6. Applied rewrites 11.7%

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

Alternative 7: 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 47.6%

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

4. Applied rewrites 7.1%

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

   1. lift-pow.f64 N/A

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

   2. pow-base-1 7.1

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

6. Applied rewrites 7.1%

   \[\leadsto \color{blue}{1} \]
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 2024308 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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