Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.5s

Alternatives: 5

Speedup: N/A×

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 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow2 99.6%

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

   4. pow2 99.6%

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

   5. pow-prod-up 99.6%

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

   6. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{4}\\ \frac{-1 - t_0}{t_0 + -1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 4.0))) (/ (- -1.0 t_0) (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 4.0);
	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) ** 4.0d0
    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), 4.0);
	return (-1.0 - t_0) / (t_0 + -1.0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 4.0)
	return (-1.0 - t_0) / (t_0 + -1.0)

Julia

function code(x)
	t_0 = tan(x) ^ 4.0
	return Float64(Float64(-1.0 - t_0) / Float64(t_0 + -1.0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 4.0;
	tmp = (-1.0 - t_0) / (t_0 + -1.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]}, N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[(t$95$0 + -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. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-1-def 99.3%

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

   6. pow2 99.3%

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

3. Applied egg-rr 99.3%

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

5. Applied egg-rr 61.3%

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

   1. *-commutative 61.3%

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

   2. neg-mul-1 61.3%

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

   3. distribute-neg-in 61.3%

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

   4. metadata-eval 61.3%

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

   5. unsub-neg 61.3%

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

   6. +-commutative 61.3%

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

   7. metadata-eval 61.3%

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

   8. sub-neg 61.3%

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

   9. difference-of-sqr--1 61.3%

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

   10. pow-sqr 61.3%

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

   11. metadata-eval 61.3%

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

   12. +-commutative 61.3%

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

7. Simplified 61.3%

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

8. Final simplification 61.3%

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

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 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow2 99.6%

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

   4. pow2 99.6%

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

   5. pow-prod-up 99.6%

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

   6. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

   1. sqrt-pow1 99.6%

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

   2. metadata-eval 99.6%

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

   3. div-sub 99.5%

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

   4. sub-neg 99.5%

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

9. Applied egg-rr 99.5%

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

    1. sub-neg 99.5%

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

    2. div-sub 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 56.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log e}{1 + \tan x \cdot \tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (log E) (+ 1.0 (* (tan x) (tan x)))))

C

double code(double x) {
	return log(((double) M_E)) / (1.0 + (tan(x) * tan(x)));
}

Java

public static double code(double x) {
	return Math.log(Math.E) / (1.0 + (Math.tan(x) * Math.tan(x)));
}

Python

def code(x):
	return math.log(math.e) / (1.0 + (math.tan(x) * math.tan(x)))

Julia

function code(x)
	return Float64(log(exp(1)) / Float64(1.0 + Float64(tan(x) * tan(x))))
end

MATLAB

function tmp = code(x)
	tmp = log(2.71828182845904523536) / (1.0 + (tan(x) * tan(x)));
end

Wolfram

code[x_] := N[(N[Log[E], $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $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. add-log-exp 98.7%

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

   2. pow2 98.7%

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

3. Applied egg-rr 98.7%

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

4. Taylor expanded in x around 0 57.4%

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

   1. exp-1-e 57.4%

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

6. Simplified 57.4%

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

7. Final simplification 57.4%

   \[\leadsto \frac{\log e}{1 + \tan x \cdot \tan x} \]

Alternative 5: 55.9% 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 99.6%

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

2. Taylor expanded in x around 0 57.0%

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

3. Final simplification 57.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023312 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))