Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.9s

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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 96.5%

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

   2. *-un-lft-identity 96.5%

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

   3. log-prod 96.5%

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

   4. metadata-eval 96.5%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, \tan x\right)\\ \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{-t_0}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (hypot 1.0 (tan x))))
   (if (or (<= (tan x) -1.0) (not (<= (tan x) 1.0)))
     (/ (- t_0) t_0)
     (/ -1.0 (- -1.0 (pow (tan x) 2.0))))))

C

double code(double x) {
	double t_0 = hypot(1.0, tan(x));
	double tmp;
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0)) {
		tmp = -t_0 / t_0;
	} else {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.hypot(1.0, Math.tan(x));
	double tmp;
	if ((Math.tan(x) <= -1.0) || !(Math.tan(x) <= 1.0)) {
		tmp = -t_0 / t_0;
	} else {
		tmp = -1.0 / (-1.0 - Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.hypot(1.0, math.tan(x))
	tmp = 0
	if (math.tan(x) <= -1.0) or not (math.tan(x) <= 1.0):
		tmp = -t_0 / t_0
	else:
		tmp = -1.0 / (-1.0 - math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x)
	t_0 = hypot(1.0, tan(x))
	tmp = 0.0
	if ((tan(x) <= -1.0) || !(tan(x) <= 1.0))
		tmp = Float64(Float64(-t_0) / t_0);
	else
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = hypot(1.0, tan(x));
	tmp = 0.0;
	if ((tan(x) <= -1.0) || ~((tan(x) <= 1.0)))
		tmp = -t_0 / t_0;
	else
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision]}, If[Or[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Not[LessEqual[N[Tan[x], $MachinePrecision], 1.0]], $MachinePrecision]], N[((-t$95$0) / t$95$0), $MachinePrecision], N[(-1.0 / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. fma-def 99.1%

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

   3. Simplified 99.1%

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

      1. add-log-exp 85.8%

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

      2. *-un-lft-identity 85.8%

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

      3. log-prod 85.8%

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

      4. metadata-eval 85.8%

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

      5. add-log-exp 99.1%

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

      6. pow2 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. +-lft-identity 99.1%

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

   7. Simplified 99.1%

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

   9. Applied egg-rr 22.3%

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

   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. 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 99.6%

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

      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/ 99.6%

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

      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. +-commutative 99.6%

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

      7. unpow2 99.6%

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

      8. fma-udef 99.7%

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

      9. fma-udef 99.7%

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

      10. neg-mul-1 99.7%

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

      11. +-commutative 99.7%

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

      12. unsub-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 75.2%

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

4. Final simplification 63.2%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 96.5%

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

   2. *-un-lft-identity 96.5%

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

   3. log-prod 96.5%

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

   4. metadata-eval 96.5%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

   1. fma-udef 99.5%

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

   2. pow2 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 4: 55.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.0 / 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. 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. 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/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. fma-udef 99.5%

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

   9. fma-udef 99.5%

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

   10. neg-mul-1 99.5%

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

   11. +-commutative 99.5%

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

   12. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 58.5%

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

7. Final simplification 58.5%

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

Alternative 5: 54.9% 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. Taylor expanded in x around 0 58.1%

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

3. Final simplification 58.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))