Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.0s

Alternatives: 6

Speedup: 0.8×

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{1 + \tan x \cdot \tan x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (- 0.0 (tan x)) (tan x) 1.0) (+ 1.0 (* (tan x) (tan x)))))

C

double code(double x) {
	return fma((0.0 - tan(x)), tan(x), 1.0) / (1.0 + (tan(x) * tan(x)));
}

Julia

function code(x)
	return Float64(fma(Float64(0.0 - tan(x)), tan(x), 1.0) / Float64(1.0 + Float64(tan(x) * tan(x))))
end

Wolfram

code[x_] := N[(N[(N[(0.0 - N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $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. sub-neg N/A

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

   2. +-commutative N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x + 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\tan x\right), \tan x, 1\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\tan x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(0 - \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
5. Add Preprocessing

Alternative 2: 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

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

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

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

   3. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

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

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

Alternative 3: 58.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(-1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $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. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

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

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

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

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

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

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

   7. pow2 N/A

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

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

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

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

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

   10. frac-2neg N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f64 N/A

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\left(1 - \tan x \cdot \tan x\right)}\right)\right)\right) \]

   14. associate--r- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\tan x \cdot \tan x}\right)\right)\right) \]

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

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

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 58.9%

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

Alternative 4: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - {\tan x}^{4}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (- 1.0 (pow (tan x) 4.0))))

C

double code(double x) {
	return 1.0 / (1.0 - pow(tan(x), 4.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 - (tan(x) ** 4.0d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.pow(Math.tan(x), 4.0));
}

Python

def code(x):
	return 1.0 / (1.0 - math.pow(math.tan(x), 4.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 - (tan(x) ^ 4.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - (tan(x) ^ 4.0));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $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. flip-+ N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

   8. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), 2\right), \left(1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)\right) \]

   11. metadata-eval N/A

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

   12. --lowering--.f64 N/A

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{{\left(1 - {\tan x}^{2}\right)}^{2}}{1 - {\tan x}^{4}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 4\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 58.1%

   \[\leadsto \frac{\color{blue}{1}}{1 - {\tan x}^{4}} \]
8. Add Preprocessing

Alternative 5: 58.8% accurate, 2.0× 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

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

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

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

   3. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

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

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 58.9%

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

8. Final simplification 58.9%

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

Alternative 6: 54.4% accurate, 411.0× 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. Simplified 54.5%

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

Reproduce

herbie shell --seed 2024156 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))