Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.3s

Alternatives: 7

Speedup: 0.8×

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 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

4. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

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

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;
	}
	return tmp;
}

Fortran

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

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;
	}
	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
	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 = -1.0;
	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;
	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], -1.0]

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-def 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%

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

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-log-exp 99.6%

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

      6. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 73.0%

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

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

   1. Initial program 99.0%

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

      1. sub-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

      4. fma-def 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. add-log-exp 92.3%

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

      2. *-un-lft-identity 92.3%

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

      3. log-prod 92.3%

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

      4. metadata-eval 92.3%

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

      5. add-log-exp 99.0%

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

      6. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. +-lft-identity 99.0%

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

   8. Simplified 99.0%

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

   10. Applied egg-rr 20.3%

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

       1. *-inverses 20.3%

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

       2. metadata-eval 20.3%

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

   12. Simplified 20.3%

       \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

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

Alternative 4: 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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.3%

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

   2. pow2 99.3%

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

4. Applied egg-rr 99.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\tan x}^{2}\right)\right)}} \]
5. Step-by-step derivation

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.3%

   \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left({\tan x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 99.5%

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

8. Simplified 99.3%

   \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left({\tan x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. expm1-udef 99.3%

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

   2. log1p-udef 99.3%

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

   3. add-exp-log 99.4%

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

   4. +-commutative 99.4%

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

   5. associate--l+ 99.4%

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

   6. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

    1. +-rgt-identity 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

   1. fma-udef 99.4%

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

   2. unpow2 99.4%

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

   3. +-commutative 99.4%

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

   4. metadata-eval 99.4%

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

   5. associate--r+ 99.3%

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

   6. sub-div 99.3%

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

   7. *-inverses 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 6: 6.4% 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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-def 99.5%

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

4. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

10. Applied egg-rr 5.8%

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

    1. *-inverses 5.8%

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

    2. metadata-eval 5.8%

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

12. Simplified 5.8%

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

13. Final simplification 5.8%

    \[\leadsto -1 \]
14. Add Preprocessing

Alternative 7: 55.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 56.5%

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

4. Final simplification 56.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))