Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 15.7s

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, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. sub-neg 99.5%

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

   4. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

7. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-/r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate--r- 99.4%

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

   5. sub0-neg 99.4%

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

   6. distribute-neg-frac 99.4%

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

   7. associate-*r/ 99.4%

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

   8. metadata-eval 99.4%

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

   9. distribute-neg-frac 99.4%

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

   10. sub0-neg 99.4%

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

   11. associate--r- 99.4%

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

   12. metadata-eval 99.4%

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

   13. associate-/r/ 99.4%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;\frac{1 + \frac{-1}{\mathsf{fma}\left(0.06666666666666667, x \cdot x, \frac{1}{x \cdot x}\right) + -0.6666666666666666}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= t_0 1.0)
     (/
      (+
       1.0
       (/
        -1.0
        (+
         (fma 0.06666666666666667 (* x x) (/ 1.0 (* x x)))
         -0.6666666666666666)))
      (+ 1.0 t_0))
     (/ (- 1.0 (* x x)) (+ 1.0 (* x x))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (t_0 <= 1.0) {
		tmp = (1.0 + (-1.0 / (fma(0.06666666666666667, (x * x), (1.0 / (x * x))) + -0.6666666666666666))) / (1.0 + t_0);
	} else {
		tmp = (1.0 - (x * x)) / (1.0 + (x * x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(fma(0.06666666666666667, Float64(x * x), Float64(1.0 / Float64(x * x))) + -0.6666666666666666))) / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], N[(N[(1.0 + N[(-1.0 / N[(N[(0.06666666666666667 * N[(x * x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * x), $MachinePrecision]), $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. Step-by-step derivation

      1. tan-quot 99.5%

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

      2. associate-*l/ 99.5%

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

      3. clear-num 99.5%

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

      4. remove-double-div 99.5%

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

      5. clear-num 99.5%

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

      6. associate-*l/ 99.5%

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

      7. tan-quot 99.6%

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

      8. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in x around 0 67.8%

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

      1. sub-neg 67.8%

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

      2. fma-def 67.8%

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

      3. unpow2 67.8%

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

      4. unpow2 67.8%

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

      5. metadata-eval 67.8%

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

   6. Simplified 67.8%

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

   if 1 < (*.f64 (tan.f64 x) (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. cancel-sign-sub-inv 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. fma-def 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in x around 0 4.4%

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

      1. +-commutative 4.4%

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

      2. unpow2 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in x around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. unpow2 9.4%

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

   9. Simplified 9.4%

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

4. Final simplification 51.4%

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

Alternative 3: 57.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 1.0 (- (pow (tan x) 2.0) -1.0))
   (/ (- 1.0 (* x x)) (+ 1.0 (* x x)))))

C

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

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 1.0:
		tmp = 1.0 / (math.pow(math.tan(x), 2.0) - -1.0)
	else:
		tmp = (1.0 - (x * x)) / (1.0 + (x * x))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.0], N[(1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * x), $MachinePrecision]), $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. Step-by-step derivation

      1. cancel-sign-sub-inv 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sub-neg 99.6%

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

      4. pow2 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around 0 67.5%

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

   if 1 < (*.f64 (tan.f64 x) (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. cancel-sign-sub-inv 99.1%

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

      2. +-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. fma-def 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in x around 0 4.4%

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

      1. +-commutative 4.4%

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

      2. unpow2 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in x around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. unsub-neg 9.4%

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

      3. unpow2 9.4%

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

   9. Simplified 9.4%

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

4. Final simplification 51.2%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (t_0 + 1.0);
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[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. fma-udef 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. tan-quot 99.3%

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

   6. associate-*r/ 99.3%

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

   7. div-sub 99.1%

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

   8. sub-neg 99.1%

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

   9. unpow2 99.1%

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

   10. associate-*r/ 99.1%

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

   11. tan-quot 99.2%

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

   12. unpow2 99.2%

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

5. Applied egg-rr 99.2%

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

   1. sub-neg 99.2%

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

   2. div-sub 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 55.0% 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 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. sub-neg 99.5%

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

   4. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in x around 0 48.6%

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

7. Final simplification 48.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))