Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.1s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

   4. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.7%

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

   6. pow2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. +-lft-identity 99.7%

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

8. Simplified 99.7%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 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. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 3: 60.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
   (+ -1.0 (/ 2.0 (pow (hypot 1.0 x) 2.0)))))

C

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

Java

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

Python

def code(x):
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 1.0:
		tmp = 1.0 / (1.0 + math.pow(math.tan(x), 2.0))
	else:
		tmp = -1.0 + (2.0 / math.pow(math.hypot(1.0, x), 2.0))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. add-log-exp 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. log-prod 99.7%

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

      4. metadata-eval 99.7%

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

      5. add-log-exp 99.7%

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

      6. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around 0 73.8%

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

   if 1 < (*.f64 (tan.f64 x) (tan.f64 x))

   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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

      4. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   6. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

      2. unpow2 99.4%

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

      3. *-inverses 99.4%

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

      4. *-inverses 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 20.9%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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]]

Derivation

1. Initial program 99.6%

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

Alternative 5: 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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. log-prod 99.5%

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

   4. metadata-eval 99.5%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

   7. add-sqr-sqrt 99.4%

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

   8. pow2 99.4%

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

   9. hypot-1-def 99.4%

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

4. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

   2. unpow2 99.4%

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

   3. hypot-undefine 99.4%

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

   4. metadata-eval 99.4%

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

   5. unpow2 99.4%

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

   6. rem-exp-log 99.4%

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

   7. log1p-undefine 99.4%

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

   8. hypot-undefine 99.3%

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

   9. metadata-eval 99.3%

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

   10. unpow2 99.3%

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

   11. rem-exp-log 99.3%

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

   12. log1p-undefine 99.3%

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

   13. rem-square-sqrt 99.5%

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

   14. log1p-undefine 99.5%

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

   15. rem-exp-log 99.6%

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

6. Simplified 99.6%

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

Alternative 6: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

   4. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.2%

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

   1. sub-neg 99.2%

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

   2. unpow2 99.2%

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

   3. *-inverses 99.2%

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

   4. *-inverses 99.2%

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

   5. metadata-eval 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. unpow2 99.2%

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

   9. hypot-undefine 99.2%

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

   10. metadata-eval 99.2%

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

   11. unpow2 99.2%

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

   12. rem-exp-log 99.2%

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

   13. log1p-undefine 99.2%

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

   14. hypot-undefine 99.2%

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

   15. metadata-eval 99.2%

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

   16. unpow2 99.2%

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

8. Simplified 99.5%

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

Alternative 7: 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 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-rgt-neg-in 99.6%

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

   4. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.7%

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

   6. pow2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. +-lft-identity 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 54.9%

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

Alternative 8: 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 99.6%

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

3. Taylor expanded in x around 0 54.6%

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))