Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.7s

Alternatives: 5

Speedup: 1.0×

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	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) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$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. --lowering--.f64 N/A

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

   2. pow2 N/A

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

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

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

   4. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. pow2 N/A

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

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

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

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

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

   4. pow2 N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. pow2 N/A

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

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

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

   11. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]
8. Add Preprocessing

Alternative 2: 60.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq 0.1:\\ \;\;\;\;\frac{-1 - \tan x}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + -1}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) 0.1) (/ (- -1.0 (tan x)) -1.0) (/ (+ (tan x) -1.0) -1.0)))

C

double code(double x) {
	double tmp;
	if (tan(x) <= 0.1) {
		tmp = (-1.0 - tan(x)) / -1.0;
	} else {
		tmp = (tan(x) + -1.0) / -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (tan(x) <= 0.1d0) then
        tmp = ((-1.0d0) - tan(x)) / (-1.0d0)
    else
        tmp = (tan(x) + (-1.0d0)) / (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= 0.1) {
		tmp = (-1.0 - Math.tan(x)) / -1.0;
	} else {
		tmp = (Math.tan(x) + -1.0) / -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= 0.1:
		tmp = (-1.0 - math.tan(x)) / -1.0
	else:
		tmp = (math.tan(x) + -1.0) / -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= 0.1)
		tmp = Float64(Float64(-1.0 - tan(x)) / -1.0);
	else
		tmp = Float64(Float64(tan(x) + -1.0) / -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= 0.1)
		tmp = (-1.0 - tan(x)) / -1.0;
	else
		tmp = (tan(x) + -1.0) / -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], 0.1], N[(N[(-1.0 - N[Tan[x], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision], N[(N[(N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < 0.10000000000000001

   1. Initial program 99.6%

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

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

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

      2. pow2 N/A

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

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

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

      4. tan-lowering-tan.f64 99.6

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

   4. Applied egg-rr 99.6%

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

      1. frac-2neg N/A

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

      2. pow2 N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. tan-quot N/A

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

      7. associate-/l* N/A

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

      8. distribute-neg-frac N/A

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

      9. distribute-frac-neg2 N/A

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

      10. frac-2neg N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

      13. +-commutative N/A

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

      14. difference-of-sqr--1 N/A

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

      15. neg-mul-1 N/A

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 71.4%

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

   if 0.10000000000000001 < (tan.f64 x)

   1. Initial program 99.1%

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

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

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

      2. pow2 N/A

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

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

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

      4. tan-lowering-tan.f64 99.1

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

   4. Applied egg-rr 99.1%

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

      1. pow2 N/A

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

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

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

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

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

      4. pow2 N/A

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. pow2 N/A

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

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

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

      11. tan-lowering-tan.f64 99.1

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

   6. Applied egg-rr 99.1%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 20.5%

       \[\leadsto \frac{\tan x + -1}{\color{blue}{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq 0.1:\\ \;\;\;\;\frac{-1 - \tan x}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + -1}{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 - {\tan x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (pow (tan x) 2.0)))

C

double code(double x) {
	return 1.0 - pow(tan(x), 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 - Math.pow(Math.tan(x), 2.0);
}

Python

def code(x):
	return 1.0 - math.pow(math.tan(x), 2.0)

Julia

function code(x)
	return Float64(1.0 - (tan(x) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (tan(x) ^ 2.0);
end

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 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. --lowering--.f64 N/A

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

   2. pow2 N/A

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

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

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

   4. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. pow2 N/A

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

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

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

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

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

   4. pow2 N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. pow2 N/A

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

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

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

   11. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 60.1%

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

10. Final simplification 60.1%

    \[\leadsto 1 - {\tan x}^{2} \]
11. Add Preprocessing

Alternative 4: 57.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + -1}{-1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $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. --lowering--.f64 N/A

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

   2. pow2 N/A

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

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

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

   4. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. pow2 N/A

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

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

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

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

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

   4. pow2 N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. pow2 N/A

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

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

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

   11. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

8. Applied egg-rr 99.3%

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

9. Taylor expanded in x around 0

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

11. Simplified 57.7%

    \[\leadsto \frac{\tan x + -1}{\color{blue}{-1}} \]
12. Add Preprocessing

Alternative 5: 55.6% 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

4. Applied egg-rr 55.8%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))