Trigonometry B

Percentage Accurate: 99.5% → 100.0%

Time: 7.2s

Alternatives: 3

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (cos (* x 2.0)))

C

double code(double x) {
	return cos((x * 2.0));
}

Fortran

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

Java

public static double code(double x) {
	return Math.cos((x * 2.0));
}

Python

def code(x):
	return math.cos((x * 2.0))

Julia

function code(x)
	return cos(Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = cos((x * 2.0));
end

Wolfram

code[x_] := N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \tan x \cdot \tan x}}{1 + \tan x \cdot \tan x} \]

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lower-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. *-rgt-identity N/A

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

    3. unpow2 N/A

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

    4. associate-*r/ N/A

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

    5. associate-*l/ N/A

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

    6. *-inverses N/A

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

    7. *-lft-identity N/A

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

    8. unpow2 N/A

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

    9. cos-sin-sum N/A

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

    10. /-rgt-identity N/A

        \[\leadsto \color{blue}{\cos \left(2 \cdot x\right)} \]

    11. metadata-eval N/A

        \[\leadsto \cos \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x\right) \]

    12. distribute-lft-neg-in N/A

        \[\leadsto \cos \color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)} \]

    13. lower-cos.f64 N/A

        \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(-2 \cdot x\right)\right)} \]

    14. distribute-lft-neg-in N/A

        \[\leadsto \cos \color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)} \]

    15. metadata-eval N/A

        \[\leadsto \cos \left(\color{blue}{2} \cdot x\right) \]

    16. lower-*.f64 100.0

        \[\leadsto \cos \color{blue}{\left(2 \cdot x\right)} \]

11. Applied rewrites 100.0%

    \[\leadsto \color{blue}{\cos \left(2 \cdot x\right)} \]

12. Final simplification 100.0%

    \[\leadsto \cos \left(x \cdot 2\right) \]
13. Add Preprocessing

Alternative 2: 56.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0) 1.0 (/ (- 1.0 (* x x)) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = (1.0 - (x * x)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 - Float64(x * x)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.6%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.5%

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

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.2%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{1 + \tan x \cdot \tan x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\tan x \cdot \tan x + 1}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\tan x \cdot \tan x} + 1} \]

      4. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{1 + {x}^{2}}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{1 - {\tan x}^{2}}{\color{blue}{x \cdot x} + 1} \]

      3. lower-fma.f64 4.2

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

   9. Applied rewrites 4.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1 - \color{blue}{{x}^{2}}}{\mathsf{fma}\left(x, x, 1\right)} \]
   11. Step-by-step derivation

       1. unpow2 N/A

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

       2. lower-*.f64 9.3

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

   12. Applied rewrites 9.3%

       \[\leadsto \frac{1 - \color{blue}{x \cdot x}}{\mathsf{fma}\left(x, x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 53.8% accurate, 428.0× 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 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))