Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

Alternatives: 7

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

\[\begin{array}{l} \\ \frac{1 - \sqrt{{\tan x}^{4}}}{1 + {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (sqrt (pow (tan x) 4.0))) (+ 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return (1.0 - sqrt(pow(tan(x), 4.0))) / (1.0 + pow(tan(x), 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - sqrt((tan(x) ** 4.0d0))) / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return (1.0 - Math.sqrt(Math.pow(Math.tan(x), 4.0))) / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return (1.0 - math.sqrt(math.pow(math.tan(x), 4.0))) / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(Float64(1.0 - sqrt((tan(x) ^ 4.0))) / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - sqrt((tan(x) ^ 4.0))) / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

code[x_] := N[(N[(1.0 - N[Sqrt[N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $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. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.5%

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

   3. pow2 99.5%

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

   4. pow2 99.5%

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

   5. pow-prod-up 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{\color{blue}{4}}}}{1 + \tan x \cdot \tan x} \]

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + {\tan x}^{2}} \]

Alternative 2: 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 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(-\left(1 - {\tan x}^{2}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(-1, {\tan x}^{2}, -1\right)}} \]
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. fma-udef 99.5%

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

   7. neg-mul-1 99.5%

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

   8. +-commutative 99.5%

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

   9. unsub-neg 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%

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

Alternative 3: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{1 - {\tan x}^{2}}\right)}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (cbrt (- 1.0 (pow (tan x) 2.0))) 3.0))

C

double code(double x) {
	return pow(cbrt((1.0 - pow(tan(x), 2.0))), 3.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt((1.0 - Math.pow(Math.tan(x), 2.0))), 3.0);
}

Julia

function code(x)
	return cbrt(Float64(1.0 - (tan(x) ^ 2.0))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Power[N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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. 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 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 59.8%

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

   1. add-cube-cbrt 59.8%

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

   2. pow3 59.8%

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

6. Applied egg-rr 59.8%

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

7. Final simplification 59.8%

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

Alternative 4: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / (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 / (1.0d0 - (tan(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 / N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 59.8%

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

   1. /-rgt-identity 59.8%

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

   2. clear-num 59.8%

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

6. Applied egg-rr 59.8%

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

7. Final simplification 59.8%

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

Alternative 5: 56.0% 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. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.5%

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

   3. pow2 99.5%

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

   4. pow2 99.5%

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

   5. pow-prod-up 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{\color{blue}{4}}}}{1 + \tan x \cdot \tan x} \]

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 55.8%

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

9. Final simplification 55.8%

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

Alternative 6: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 59.8%

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

5. Final simplification 59.8%

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

Alternative 7: 55.6% 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 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{1 - \sqrt{{\tan x}^{4}}}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

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%

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

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. Taylor expanded in x around 0 55.5%

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

11. Final simplification 55.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))