Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.7s

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, 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 97.5%

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

   2. *-un-lft-identity 97.5%

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

   3. log-prod 97.5%

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

   4. metadata-eval 97.5%

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

   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%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

Alternative 2: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t\_0 \leq 1.005:\\ \;\;\;\;\frac{1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {x}^{2}}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= t_0 1.005)
     (/ 1.0 (+ 1.0 t_0))
     (/ (- 1.0 (pow x 2.0)) (fma x x 1.0)))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (t_0 <= 1.005) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = (1.0 - pow(x, 2.0)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (t_0 <= 1.005)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.005], N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1.0049999999999999

   1. Initial program 99.5%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 73.3%

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

   if 1.0049999999999999 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.4%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 4.2%

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

   4. Taylor expanded in x around 0 9.9%

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

      1. +-commutative 9.9%

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

      2. unpow2 9.9%

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

      3. fma-define 9.9%

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

   6. Simplified 9.9%

      \[\leadsto \frac{1 - {x}^{2}}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.5%

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

   6. add-sqr-sqrt 99.2%

      \[\leadsto 0 + \frac{1 - \tan x \cdot \tan x}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   7. pow2 99.2%

      \[\leadsto 0 + \frac{1 - \color{blue}{{\tan x}^{2}}}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}} \]

   8. pow2 99.2%

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

   9. hypot-1-def 99.3%

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

4. Applied egg-rr 99.3%

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

   1. +-lft-identity 99.3%

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

   2. unpow2 99.3%

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

   3. hypot-undefine 99.3%

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

   4. metadata-eval 99.3%

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

   5. unpow2 99.3%

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

   6. rem-exp-log 99.2%

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

   7. log1p-undefine 99.2%

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

   8. hypot-undefine 99.2%

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

   9. metadata-eval 99.2%

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

   10. unpow2 99.2%

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

   11. rem-exp-log 99.1%

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

   12. log1p-undefine 99.1%

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

   13. rem-square-sqrt 99.3%

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

   14. log1p-undefine 99.3%

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

   15. rem-exp-log 99.5%

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

6. Simplified 99.5%

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

Alternative 4: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - {\tan x}^{2}\right) \cdot \mathsf{expm1}\left(\log 2\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (- 1.0 (pow (tan x) 2.0)) (expm1 (log 2.0))))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) * expm1(log(2.0));
}

Java

public static double code(double x) {
	return (1.0 - Math.pow(Math.tan(x), 2.0)) * Math.expm1(Math.log(2.0));
}

Python

def code(x):
	return (1.0 - math.pow(math.tan(x), 2.0)) * math.expm1(math.log(2.0))

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) * expm1(log(2.0)))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[2.0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

   3. add-sqr-sqrt 99.2%

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

   4. pow2 99.2%

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

   5. hypot-1-def 99.2%

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

   6. pow2 99.2%

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

4. Applied egg-rr 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 98.4%

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

   3. pow-flip 98.4%

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

   4. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

   1. expm1-define 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in x around 0 59.1%

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

10. Final simplification 59.1%

    \[\leadsto \left(1 - {\tan x}^{2}\right) \cdot \mathsf{expm1}\left(\log 2\right) \]
11. Add Preprocessing

Alternative 5: 55.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 55.1%

   \[\leadsto \frac{\color{blue}{1}}{1 + \tan x \cdot \tan x} \]
4. Add Preprocessing

Alternative 6: 55.0% 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 97.5%

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

   2. *-un-lft-identity 97.5%

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

   3. log-prod 97.5%

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

   4. metadata-eval 97.5%

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

   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%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in x around 0 54.8%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))