Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 12.8s

Alternatives: 5

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 99.6%

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

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], N[Power[-1.0, 1/3], $MachinePrecision], If[LessEqual[N[Tan[x], $MachinePrecision], 1.0], N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[-1.0, 1/3], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

      4. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

   7. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{\sqrt[3]{-1} \cdot \frac{\mathsf{hypot}\left(1, \tan x\right)}{\mathsf{hypot}\left(1, \tan x\right)}} \]
   8. Step-by-step derivation

      1. *-inverses 20.7%

         \[\leadsto \sqrt[3]{-1} \cdot \color{blue}{1} \]

      2. *-rgt-identity 20.7%

         \[\leadsto \color{blue}{\sqrt[3]{-1}} \]

   9. Simplified 20.7%

      \[\leadsto \color{blue}{\sqrt[3]{-1}} \]

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-log-exp 99.7%

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

      6. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   7. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. pow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around 0 73.1%

       \[\leadsto \frac{\color{blue}{1}}{{\tan x}^{2} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;\sqrt[3]{-1}\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-1}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t_0}{1 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-def 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. Step-by-step derivation

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. pow2 99.6%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 4: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. frac-2neg 99.6%

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

   2. div-inv 99.4%

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

   3. pow2 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-neg-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. neg-mul-1 99.5%

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

   8. fma-def 99.5%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 58.2%

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

5. Final simplification 58.2%

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

Alternative 5: 55.5% 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. 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-def 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. Step-by-step derivation

   1. add-log-exp 98.8%

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

   2. *-un-lft-identity 98.8%

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

   3. log-prod 98.8%

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

   4. metadata-eval 98.8%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

7. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. pow2 99.6%

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

9. Applied egg-rr 99.6%

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

10. Taylor expanded in x around 0 53.7%

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

11. Final simplification 53.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))