Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.9s

Alternatives: 6

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, 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 99.4%

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

   1. add-log-exp 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. log-prod 98.2%

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

   4. metadata-eval 98.2%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

5. Simplified 99.4%

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

   1. pow2 99.4%

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

   2. pow2 99.4%

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

   3. metadata-eval 99.4%

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

   4. sqrt-pow2 47.7%

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

   5. expm1-log1p-u 47.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)\right)} \]

   6. expm1-udef 47.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)} - 1} \]

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 55.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around 0 57.7%

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

   1. expm1-log1p-u 57.7%

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

   2. expm1-udef 57.7%

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

   3. pow2 57.7%

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

   4. add-sqr-sqrt 57.7%

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

   5. pow2 57.7%

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

   6. pow2 57.7%

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

   7. hypot-1-def 57.7%

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

4. Applied egg-rr 57.7%

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

   1. expm1-def 57.7%

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

   2. expm1-log1p 57.7%

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

6. Simplified 57.7%

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

7. Final simplification 57.7%

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

Alternative 3: 58.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. frac-2neg 99.4%

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

   2. div-inv 99.3%

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

   3. pow2 99.3%

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

   4. +-commutative 99.3%

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

   5. distribute-neg-in 99.3%

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

   6. neg-mul-1 99.3%

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

   7. metadata-eval 99.3%

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

   8. fma-def 99.3%

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

   9. pow2 99.3%

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

3. Applied egg-rr 99.3%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. neg-sub0 99.4%

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

   4. associate--r- 99.4%

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

   5. metadata-eval 99.4%

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

   6. +-commutative 99.4%

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

   7. unpow2 99.4%

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

   8. fma-def 99.4%

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

   9. fma-udef 99.4%

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

   10. neg-mul-1 99.4%

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

   11. +-commutative 99.4%

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

   12. unsub-neg 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in x around 0 62.0%

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

7. Final simplification 62.0%

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

Alternative 4: 55.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around 0 57.7%

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

   1. expm1-log1p-u 57.7%

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

   2. expm1-udef 57.7%

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

   3. pow2 57.7%

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

   4. add-sqr-sqrt 57.7%

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

   5. pow2 57.7%

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

   6. pow2 57.7%

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

   7. hypot-1-def 57.7%

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

4. Applied egg-rr 57.7%

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

   1. expm1-def 57.7%

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

   2. expm1-log1p 57.7%

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

   3. unpow2 57.7%

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

   4. associate-/r* 57.7%

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

   5. *-lft-identity 57.7%

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

   6. associate-*l/ 57.7%

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

   7. unpow-1 57.7%

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

   8. unpow-1 57.7%

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

   9. pow-sqr 57.7%

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

   10. metadata-eval 57.7%

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

6. Simplified 57.7%

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

7. Final simplification 57.7%

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

Alternative 5: 55.0% 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 99.4%

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

   1. add-log-exp 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. log-prod 98.2%

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

   4. metadata-eval 98.2%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

5. Simplified 99.4%

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

   1. pow2 99.4%

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

   2. pow2 99.4%

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

   3. metadata-eval 99.4%

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

   4. sqrt-pow2 47.7%

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

   5. expm1-log1p-u 47.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)\right)} \]

   6. expm1-udef 47.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)} - 1} \]

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around 0 57.7%

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

11. Final simplification 57.7%

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

Alternative 6: 54.7% 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 99.4%

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

   1. add-log-exp 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. log-prod 98.2%

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

   4. metadata-eval 98.2%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

5. Simplified 99.4%

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

   1. pow2 99.4%

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

   2. pow2 99.4%

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

   3. metadata-eval 99.4%

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

   4. sqrt-pow2 47.7%

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

   5. expm1-log1p-u 47.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)\right)} \]

   6. expm1-udef 47.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \tan x \cdot \tan x}{1 + {\left(\sqrt{\tan x}\right)}^{4}}\right)} - 1} \]

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around 0 57.4%

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

11. Final simplification 57.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))