Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.0s

Alternatives: 9

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 99.5%

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

   1. /-rgt-identity 99.5%

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

   2. div-sub 99.3%

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

   3. sub-neg 99.3%

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

   4. +-commutative 99.3%

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

   5. neg-sub0 99.3%

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

   6. associate-+l- 99.3%

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

   7. sub0-neg 99.3%

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

   8. neg-mul-1 99.3%

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

   9. *-commutative 99.3%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

   2. log1p-def 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

   3. expm1-udef 99.3%

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

   4. add-exp-log 99.5%

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

   5. +-commutative 99.5%

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

   6. associate--l+ 99.5%

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

   7. pow2 99.5%

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

   8. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-rgt-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = tan(x) / (1.0 / 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) / (1.0d0 / 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) / (1.0 / Math.tan(x));
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) / (1.0 / 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[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. frac-2neg 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-udef 99.5%

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

   4. fma-udef 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. sub-neg 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. flip-- 99.4%

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

   11. associate-/l/ 99.3%

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

3. Applied egg-rr 99.5%

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

   1. distribute-neg-frac 99.5%

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

   2. distribute-neg-in 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-commutative 99.5%

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

   5. sub-neg 99.5%

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

   6. +-commutative 99.5%

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

5. Simplified 99.5%

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

   1. unpow2 99.5%

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

   2. tan-quot 99.4%

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

   3. associate-*r/ 99.4%

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

   4. associate-/l* 99.4%

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

   5. clear-num 99.4%

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

   6. tan-quot 99.5%

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

7. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

   2. tan-quot 99.4%

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

   3. associate-*r/ 99.4%

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

   4. associate-/l* 99.4%

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

   5. clear-num 99.4%

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

   6. tan-quot 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 60.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.0) {
		tmp = -1.0 / (1.0 + (-1.0 + (-1.0 - pow(tan(x), 2.0))));
	} else {
		tmp = -1.0;
	}
	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) / (1.0d0 + ((-1.0d0) + ((-1.0d0) - (tan(x) ** 2.0d0))))
    else
        tmp = -1.0d0
    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 / (1.0 + (-1.0 + (-1.0 - Math.pow(Math.tan(x), 2.0))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 1.0)
		tmp = -1.0 / (1.0 + (-1.0 + (-1.0 - (tan(x) ^ 2.0))));
	else
		tmp = -1.0;
	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[(1.0 + N[(-1.0 + N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

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. /-rgt-identity 99.6%

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

      2. div-sub 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. neg-sub0 99.4%

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

      6. associate-+l- 99.4%

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

      7. sub0-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-commutative 99.4%

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

   3. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 99.6%

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

      4. add-exp-log 99.6%

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

      5. +-commutative 99.6%

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

      6. associate--l+ 99.6%

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

      7. pow2 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 70.4%

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

      1. expm1-log1p-u 70.4%

         \[\leadsto \frac{-1}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\tan x}^{2}\right)\right)}} \]

      2. log1p-def 70.4%

         \[\leadsto \frac{-1}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + {\tan x}^{2}\right)}\right)} \]

      3. expm1-udef 70.4%

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

      4. add-exp-log 70.4%

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

      5. associate--r- 70.4%

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

      6. +-commutative 70.4%

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

   10. Applied egg-rr 70.4%

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

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

   1. Initial program 99.2%

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

      1. /-rgt-identity 99.2%

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

      2. div-sub 99.0%

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

      3. sub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. neg-sub0 99.0%

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

      6. associate-+l- 99.0%

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

      7. sub0-neg 99.0%

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

      8. neg-mul-1 99.0%

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

      9. *-commutative 99.0%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 98.7%

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

      4. add-exp-log 99.3%

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

      5. +-commutative 99.3%

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

      6. associate--l+ 99.3%

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

      7. pow2 99.3%

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

      8. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-rgt-identity 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 1.6%

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

   10. Applied egg-rr 17.3%

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

   11. Taylor expanded in x around 0 20.7%

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

4. Final simplification 58.0%

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

Alternative 4: 60.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.0) {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	} else {
		tmp = -1.0;
	}
	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) / ((-1.0d0) - (tan(x) ** 2.0d0))
    else
        tmp = -1.0d0
    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 / (-1.0 - Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 1.0)
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	else
		tmp = -1.0;
	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[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

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. /-rgt-identity 99.6%

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

      2. div-sub 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. neg-sub0 99.4%

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

      6. associate-+l- 99.4%

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

      7. sub0-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-commutative 99.4%

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

   3. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 99.6%

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

      4. add-exp-log 99.6%

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

      5. +-commutative 99.6%

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

      6. associate--l+ 99.6%

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

      7. pow2 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 70.4%

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

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

   1. Initial program 99.2%

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

      1. /-rgt-identity 99.2%

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

      2. div-sub 99.0%

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

      3. sub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. neg-sub0 99.0%

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

      6. associate-+l- 99.0%

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

      7. sub0-neg 99.0%

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

      8. neg-mul-1 99.0%

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

      9. *-commutative 99.0%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 98.7%

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

      4. add-exp-log 99.3%

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

      5. +-commutative 99.3%

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

      6. associate--l+ 99.3%

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

      7. pow2 99.3%

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

      8. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-rgt-identity 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 1.6%

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

   10. Applied egg-rr 17.3%

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

   11. Taylor expanded in x around 0 20.7%

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

4. Final simplification 58.0%

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

Alternative 5: 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 99.5%

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

   1. frac-2neg 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-udef 99.5%

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

   4. fma-udef 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. sub-neg 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. flip-- 99.4%

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

   11. associate-/l/ 99.3%

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

3. Applied egg-rr 99.5%

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

   1. distribute-neg-frac 99.5%

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

   2. distribute-neg-in 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-commutative 99.5%

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

   5. sub-neg 99.5%

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

   6. +-commutative 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{1 + \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (pow (tan x) 2.0))
  (+ 1.0 (/ (tan x) (+ (* x -0.3333333333333333) (/ 1.0 x))))))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / (1.0 + (tan(x) / ((x * -0.3333333333333333) + (1.0 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) ** 2.0d0)) / (1.0d0 + (tan(x) / ((x * (-0.3333333333333333d0)) + (1.0d0 / x))))
end function

Java

public static double code(double x) {
	return (1.0 - Math.pow(Math.tan(x), 2.0)) / (1.0 + (Math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x))));
}

Python

def code(x):
	return (1.0 - math.pow(math.tan(x), 2.0)) / (1.0 + (math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x))))

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / Float64(1.0 + Float64(tan(x) / Float64(Float64(x * -0.3333333333333333) + Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) ^ 2.0)) / (1.0 + (tan(x) / ((x * -0.3333333333333333) + (1.0 / x))));
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] / N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. frac-2neg 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-udef 99.5%

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

   4. fma-udef 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. +-commutative 99.5%

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

   8. sub-neg 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. flip-- 99.4%

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

   11. associate-/l/ 99.3%

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

3. Applied egg-rr 99.5%

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

   1. distribute-neg-frac 99.5%

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

   2. distribute-neg-in 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-commutative 99.5%

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

   5. sub-neg 99.5%

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

   6. +-commutative 99.5%

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

5. Simplified 99.5%

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

   1. unpow2 99.5%

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

   2. tan-quot 99.4%

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

   3. associate-*r/ 99.4%

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

   4. associate-/l* 99.4%

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

   5. clear-num 99.4%

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

   6. tan-quot 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in x around 0 57.4%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \frac{\tan x}{\color{blue}{-0.3333333333333333 \cdot x + \frac{1}{x}}}} \]

9. Final simplification 57.4%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}} \]

Alternative 7: 59.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * 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. Step-by-step derivation

   1. /-rgt-identity 99.5%

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

   2. div-sub 99.3%

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

   3. sub-neg 99.3%

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

   4. +-commutative 99.3%

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

   5. neg-sub0 99.3%

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

   6. associate-+l- 99.3%

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

   7. sub0-neg 99.3%

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

   8. neg-mul-1 99.3%

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

   9. *-commutative 99.3%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

   2. log1p-def 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

   3. expm1-udef 99.3%

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

   4. add-exp-log 99.5%

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

   5. +-commutative 99.5%

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

   6. associate--l+ 99.5%

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

   7. pow2 99.5%

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

   8. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-rgt-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 57.2%

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

9. Final simplification 57.2%

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

Alternative 8: 60.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], -1.0, If[LessEqual[N[Tan[x], $MachinePrecision], 1.0], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.2%

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

      1. /-rgt-identity 99.2%

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

      2. div-sub 99.0%

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

      3. sub-neg 99.0%

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

      4. +-commutative 99.0%

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

      5. neg-sub0 99.0%

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

      6. associate-+l- 99.0%

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

      7. sub0-neg 99.0%

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

      8. neg-mul-1 99.0%

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

      9. *-commutative 99.0%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 98.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 98.7%

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

      4. add-exp-log 99.3%

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

      5. +-commutative 99.3%

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

      6. associate--l+ 99.3%

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

      7. pow2 99.3%

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

      8. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. +-rgt-identity 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 1.6%

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

   10. Applied egg-rr 17.3%

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

   11. Taylor expanded in x around 0 20.7%

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

   if -1 < (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. /-rgt-identity 99.6%

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

      2. div-sub 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. neg-sub0 99.4%

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

      6. associate-+l- 99.4%

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

      7. sub0-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-commutative 99.4%

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

   3. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

      2. log1p-def 99.6%

         \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

      3. expm1-udef 99.6%

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

      4. add-exp-log 99.6%

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

      5. +-commutative 99.6%

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

      6. associate--l+ 99.6%

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

      7. pow2 99.6%

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

      8. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 69.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 6.5% 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. Step-by-step derivation

   1. /-rgt-identity 99.5%

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

   2. div-sub 99.3%

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

   3. sub-neg 99.3%

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

   4. +-commutative 99.3%

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

   5. neg-sub0 99.3%

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

   6. associate-+l- 99.3%

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

   7. sub0-neg 99.3%

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

   8. neg-mul-1 99.3%

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

   9. *-commutative 99.3%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x \cdot \tan x\right)\right)}} \]

   2. log1p-def 99.3%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + \tan x \cdot \tan x\right)}\right)} \]

   3. expm1-udef 99.3%

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

   4. add-exp-log 99.5%

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

   5. +-commutative 99.5%

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

   6. associate--l+ 99.5%

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

   7. pow2 99.5%

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

   8. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-rgt-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 53.2%

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

10. Applied egg-rr 5.5%

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

11. Taylor expanded in x around 0 6.4%

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

12. Final simplification 6.4%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))