Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 20.8s

Alternatives: 4

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 99.6%

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

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

      \[\leadsto \frac{1 - \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)}{1 + \tan x \cdot \tan x} \]

   5. add-log-exp 99.6%

      \[\leadsto \frac{1 - \left(0 + \color{blue}{\tan x \cdot \tan x}\right)}{1 + \tan x \cdot \tan x} \]

   6. pow2 99.6%

      \[\leadsto \frac{1 - \left(0 + \color{blue}{{\tan x}^{2}}\right)}{1 + \tan x \cdot \tan x} \]

3. Applied egg-rr 99.6%

   \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. +-lft-identity 99.6%

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

5. Simplified 99.6%

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

   1. pow2 99.6%

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

   2. pow2 99.6%

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

   3. metadata-eval 99.6%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\tan x}^{\color{blue}{\left(0.3333333333333333 \cdot 6\right)}}} \]

   4. pow-pow 44.6%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{{\left({\tan x}^{0.3333333333333333}\right)}^{6}}} \]

   5. pow1/3 98.9%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\sqrt[3]{\tan x}\right)}}^{6}} \]

   6. div-sub 98.9%

      \[\leadsto \color{blue}{\frac{1}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}} - \frac{\tan x \cdot \tan x}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}} \]

   7. sub-neg 98.9%

      \[\leadsto \color{blue}{\frac{1}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}} + \left(-\frac{\tan x \cdot \tan x}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}\right)} \]

   8. pow1/3 44.4%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\tan x}^{0.3333333333333333}\right)}}^{6}} + \left(-\frac{\tan x \cdot \tan x}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}\right) \]

   9. pow-pow 98.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\tan x}^{\left(0.3333333333333333 \cdot 6\right)}}} + \left(-\frac{\tan x \cdot \tan x}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}\right) \]

   10. metadata-eval 98.8%

       \[\leadsto \frac{1}{1 + {\tan x}^{\color{blue}{2}}} + \left(-\frac{\tan x \cdot \tan x}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}\right) \]

   11. pow2 98.8%

       \[\leadsto \frac{1}{1 + {\tan x}^{2}} + \left(-\frac{\color{blue}{{\tan x}^{2}}}{1 + {\left(\sqrt[3]{\tan x}\right)}^{6}}\right) \]

7. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

   2. div-sub 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{1}{x \cdot x} + \left(\left(x \cdot x\right) \cdot 0.06666666666666667 - 0.6666666666666666\right)}}{1 + \tan x \cdot \tan x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (+ -1.0 (/ 2.0 (* x x)))
   (/
    (+
     1.0
     (/
      -1.0
      (+
       (/ 1.0 (* x x))
       (- (* (* x x) 0.06666666666666667) 0.6666666666666666))))
    (+ 1.0 (* (tan x) (tan x))))))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 + (2.0 / (x * x));
	} else {
		tmp = (1.0 + (-1.0 / ((1.0 / (x * x)) + (((x * x) * 0.06666666666666667) - 0.6666666666666666)))) / (1.0 + (tan(x) * tan(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((tan(x) <= (-1.0d0)) .or. (.not. (tan(x) <= 1.0d0))) then
        tmp = (-1.0d0) + (2.0d0 / (x * x))
    else
        tmp = (1.0d0 + ((-1.0d0) / ((1.0d0 / (x * x)) + (((x * x) * 0.06666666666666667d0) - 0.6666666666666666d0)))) / (1.0d0 + (tan(x) * tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 + (2.0 / (x * x));
	} else {
		tmp = (1.0 + (-1.0 / ((1.0 / (x * x)) + (((x * x) * 0.06666666666666667) - 0.6666666666666666)))) / (1.0 + (Math.tan(x) * Math.tan(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 + (2.0 / (x * x))
	else:
		tmp = (1.0 + (-1.0 / ((1.0 / (x * x)) + (((x * x) * 0.06666666666666667) - 0.6666666666666666)))) / (1.0 + (math.tan(x) * math.tan(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(Float64(1.0 / Float64(x * x)) + Float64(Float64(Float64(x * x) * 0.06666666666666667) - 0.6666666666666666)))) / Float64(1.0 + Float64(tan(x) * tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 + (2.0 / (x * x));
	else
		tmp = (1.0 + (-1.0 / ((1.0 / (x * x)) + (((x * x) * 0.06666666666666667) - 0.6666666666666666)))) / (1.0 + (tan(x) * tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-1.0 / N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * 0.06666666666666667), $MachinePrecision] - 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.1%

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

      1. add-log-exp 95.2%

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

      2. *-un-lft-identity 95.2%

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

      3. log-prod 95.2%

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

      4. metadata-eval 95.2%

         \[\leadsto \frac{1 - \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)}{1 + \tan x \cdot \tan x} \]

      5. add-log-exp 99.1%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\tan x \cdot \tan x}\right)}{1 + \tan x \cdot \tan x} \]

      6. pow2 99.1%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{{\tan x}^{2}}\right)}{1 + \tan x \cdot \tan x} \]

   3. Applied egg-rr 99.1%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \tan x \cdot \tan x} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 4.2%

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

      1. unpow2 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in x around 0 9.9%

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

       1. unpow2 9.9%

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

   11. Simplified 9.9%

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

   12. Taylor expanded in x around inf 20.6%

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

       1. sub-neg 20.6%

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

       2. associate-*r/ 20.6%

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

       3. metadata-eval 20.6%

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

       4. unpow2 20.6%

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

       5. metadata-eval 20.6%

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

   14. Simplified 20.6%

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

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

   1. Initial program 99.7%

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

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

         \[\leadsto \frac{1 - \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)}{1 + \tan x \cdot \tan x} \]

      5. add-log-exp 99.7%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\tan x \cdot \tan x}\right)}{1 + \tan x \cdot \tan x} \]

      6. pow2 99.7%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{{\tan x}^{2}}\right)}{1 + \tan x \cdot \tan x} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \tan x \cdot \tan x} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.7%

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

   5. Simplified 99.7%

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

      1. pow2 99.7%

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

      2. tan-quot 99.6%

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

      3. tan-quot 99.6%

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

      4. frac-times 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. clear-num 99.6%

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

      8. clear-num 99.6%

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

      9. unpow2 99.6%

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

      10. unpow2 99.6%

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

      11. frac-times 99.6%

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

      12. tan-quot 99.7%

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

      13. tan-quot 99.7%

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

      14. pow2 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 72.4%

      \[\leadsto \frac{1 - \frac{1}{\color{blue}{\left(\frac{1}{{x}^{2}} + 0.06666666666666667 \cdot {x}^{2}\right) - 0.6666666666666666}}}{1 + \tan x \cdot \tan x} \]
   9. Step-by-step derivation

      1. associate--l+ 72.4%

         \[\leadsto \frac{1 - \frac{1}{\color{blue}{\frac{1}{{x}^{2}} + \left(0.06666666666666667 \cdot {x}^{2} - 0.6666666666666666\right)}}}{1 + \tan x \cdot \tan x} \]

      2. unpow2 72.4%

         \[\leadsto \frac{1 - \frac{1}{\frac{1}{\color{blue}{x \cdot x}} + \left(0.06666666666666667 \cdot {x}^{2} - 0.6666666666666666\right)}}{1 + \tan x \cdot \tan x} \]

      3. *-commutative 72.4%

         \[\leadsto \frac{1 - \frac{1}{\frac{1}{x \cdot x} + \left(\color{blue}{{x}^{2} \cdot 0.06666666666666667} - 0.6666666666666666\right)}}{1 + \tan x \cdot \tan x} \]

      4. unpow2 72.4%

         \[\leadsto \frac{1 - \frac{1}{\frac{1}{x \cdot x} + \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.06666666666666667 - 0.6666666666666666\right)}}{1 + \tan x \cdot \tan x} \]

   10. Simplified 72.4%

       \[\leadsto \frac{1 - \frac{1}{\color{blue}{\frac{1}{x \cdot x} + \left(\left(x \cdot x\right) \cdot 0.06666666666666667 - 0.6666666666666666\right)}}}{1 + \tan x \cdot \tan x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{1}{x \cdot x} + \left(\left(x \cdot x\right) \cdot 0.06666666666666667 - 0.6666666666666666\right)}}{1 + \tan x \cdot \tan x}\\ \end{array} \]

Alternative 3: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
   (+ -1.0 (/ 2.0 (* x x)))
   (/ -1.0 (- -1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -1.0 + (2.0 / (x * x));
	} else {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((tan(x) <= (-1.0d0)) .or. (.not. (tan(x) <= 1.0d0))) then
        tmp = (-1.0d0) + (2.0d0 / (x * x))
    else
        tmp = (-1.0d0) / ((-1.0d0) - (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -1.0 + (2.0 / (x * x));
	} else {
		tmp = -1.0 / (-1.0 - Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -1.0 + (2.0 / (x * x))
	else:
		tmp = -1.0 / (-1.0 - math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(x * x)));
	else
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -1.0 + (2.0 / (x * x));
	else
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.1%

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

      1. add-log-exp 95.2%

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

      2. *-un-lft-identity 95.2%

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

      3. log-prod 95.2%

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

      4. metadata-eval 95.2%

         \[\leadsto \frac{1 - \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)}{1 + \tan x \cdot \tan x} \]

      5. add-log-exp 99.1%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{\tan x \cdot \tan x}\right)}{1 + \tan x \cdot \tan x} \]

      6. pow2 99.1%

         \[\leadsto \frac{1 - \left(0 + \color{blue}{{\tan x}^{2}}\right)}{1 + \tan x \cdot \tan x} \]

   3. Applied egg-rr 99.1%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \tan x \cdot \tan x} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 4.2%

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

      1. unpow2 4.2%

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

   8. Simplified 4.2%

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

   9. Taylor expanded in x around 0 9.9%

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

       1. unpow2 9.9%

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

   11. Simplified 9.9%

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

   12. Taylor expanded in x around inf 20.6%

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

       1. sub-neg 20.6%

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

       2. associate-*r/ 20.6%

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

       3. metadata-eval 20.6%

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

       4. unpow2 20.6%

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

       5. metadata-eval 20.6%

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

   14. Simplified 20.6%

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

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

   1. Initial program 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. pow2 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. neg-mul-1 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-def 99.6%

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

      9. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. neg-sub0 99.7%

         \[\leadsto \frac{\color{blue}{0 - \left(1 - {\tan x}^{2}\right)}}{\mathsf{fma}\left(-1, {\tan x}^{2}, -1\right)} \]

      4. associate--r- 99.7%

         \[\leadsto \frac{\color{blue}{\left(0 - 1\right) + {\tan x}^{2}}}{\mathsf{fma}\left(-1, {\tan x}^{2}, -1\right)} \]

      5. metadata-eval 99.7%

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

      6. +-commutative 99.7%

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

      7. unpow2 99.7%

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

      8. fma-udef 99.7%

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

      9. fma-udef 99.7%

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

      10. neg-mul-1 99.7%

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

      11. +-commutative 99.7%

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

      12. unsub-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 71.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;-1 + \frac{2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \end{array} \]

Alternative 4: 54.6% 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 99.6%

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

2. Taylor expanded in x around 0 58.0%

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

3. Final simplification 58.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))