Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.8s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $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. +-commutative 99.6%

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

   2. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.25:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.25)
   (/ -1.0 (- -1.0 (pow (tan x) 2.0)))
   (/ (- 1.0 (* x x)) (+ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.25) {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	} 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)) <= 1.25d0) then
        tmp = (-1.0d0) / ((-1.0d0) - (tan(x) ** 2.0d0))
    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)) <= 1.25) {
		tmp = -1.0 / (-1.0 - Math.pow(Math.tan(x), 2.0));
	} 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)) <= 1.25:
		tmp = -1.0 / (-1.0 - math.pow(math.tan(x), 2.0))
	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)) <= 1.25)
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	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)) <= 1.25)
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	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], 1.25], N[(-1.0 / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $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)) < 1.25

   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. fma-udef 99.7%

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

      7. neg-mul-1 99.7%

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

      8. +-commutative 99.7%

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

      9. unsub-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 74.8%

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

   if 1.25 < (*.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. add-cbrt-cube 98.7%

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

      2. pow1/3 98.3%

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

      3. pow3 98.2%

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

      4. add-sqr-sqrt 98.2%

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

      5. pow2 98.2%

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\left({\color{blue}{\left({\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333}} \]

      6. pow-pow 98.4%

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\color{blue}{\left({\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333}} \]

      7. hypot-1-def 98.3%

         \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\left({\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}} \]

      8. metadata-eval 98.3%

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

   3. Applied egg-rr 98.3%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{{\left({\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{6}\right)}^{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. unpow1/3 98.7%

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

   5. Simplified 98.7%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{6}}}} \]
   6. Step-by-step derivation

      1. add-log-exp 94.2%

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

      2. *-un-lft-identity 94.2%

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

      3. log-prod 94.2%

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

      4. metadata-eval 94.2%

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

      5. add-log-exp 99.3%

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

      6. pow2 99.3%

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

   7. Applied egg-rr 98.7%

      \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{6}}} \]
   8. Step-by-step derivation

      1. +-lft-identity 99.3%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around 0 4.2%

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

       1. +-commutative 4.2%

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

       2. unpow2 4.2%

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

   12. Simplified 4.2%

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

   13. Taylor expanded in x around 0 8.9%

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

       1. unpow2 8.9%

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

   15. Simplified 8.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.25:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(t$95$0 + -1.0), $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. frac-2neg 99.6%

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

   2. div-inv 99.5%

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

   3. pow2 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. neg-mul-1 99.5%

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

   7. metadata-eval 99.5%

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

   8. fma-def 99.5%

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

   9. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. neg-sub0 99.6%

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

   4. associate--r- 99.6%

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

   5. metadata-eval 99.6%

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

   6. fma-udef 99.6%

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

   7. neg-mul-1 99.6%

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

   8. +-commutative 99.6%

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

   9. unsub-neg 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. flip-+ 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow2 99.5%

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

   4. metadata-eval 99.5%

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

   5. pow2 99.5%

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

   6. pow2 99.5%

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

   7. pow-prod-up 99.4%

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

   8. metadata-eval 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*l/ 99.4%

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

   2. unpow2 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0 61.3%

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

7. Final simplification 61.3%

   \[\leadsto \frac{1}{1 - {\tan x}^{4}} \]

Alternative 5: 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. Step-by-step derivation

   1. frac-2neg 99.6%

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

   2. div-inv 99.5%

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

   3. pow2 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. neg-mul-1 99.5%

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

   7. metadata-eval 99.5%

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

   8. fma-def 99.5%

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

   9. pow2 99.5%

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

3. Applied egg-rr 99.5%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. neg-sub0 99.6%

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

   4. associate--r- 99.6%

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

   5. metadata-eval 99.6%

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

   6. fma-udef 99.6%

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

   7. neg-mul-1 99.6%

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

   8. +-commutative 99.6%

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

   9. unsub-neg 99.6%

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

5. Simplified 99.6%

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

   1. add-cbrt-cube 99.4%

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

   2. pow3 99.4%

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

   3. pow-pow 99.4%

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

   4. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

   \[\leadsto \frac{-1 + \color{blue}{\sqrt[3]{{\tan x}^{6}}}}{-1 - {\tan x}^{2}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 + \sqrt[3]{{\tan x}^{6}}\right)\right)}}{-1 - {\tan x}^{2}} \]

   2. expm1-udef 99.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-1 + \sqrt[3]{{\tan x}^{6}}\right)} - 1}}{-1 - {\tan x}^{2}} \]

   3. +-commutative 99.3%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\tan x}^{6}} + -1}\right)} - 1}{-1 - {\tan x}^{2}} \]

   4. pow1/3 99.4%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left({\tan x}^{6}\right)}^{0.3333333333333333}} + -1\right)} - 1}{-1 - {\tan x}^{2}} \]

   5. pow-pow 99.4%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{{\tan x}^{\left(6 \cdot 0.3333333333333333\right)}} + -1\right)} - 1}{-1 - {\tan x}^{2}} \]

   6. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\tan x}^{2} + -1\right)} - 1}}{-1 - {\tan x}^{2}} \]
10. Step-by-step derivation

    1. expm1-def 99.4%

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

    2. expm1-log1p 99.6%

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

    3. unpow2 99.6%

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

    4. fma-udef 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in x around 0 58.2%

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

13. Final simplification 58.2%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023227 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))