Trigonometry B

Percentage Accurate: 99.5% → 98.3%

Time: 12.4s

Alternatives: 6

Speedup: N/A×

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: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{\log \left(e^{1 - t\_0}\right)}{1 + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = log(exp((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[Log[N[Exp[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $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 99.6%

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

   2. pow2 99.6%

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

4. Applied egg-rr 99.6%

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

   1. add-log-exp 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. log-prod 99.5%

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

   4. metadata-eval 99.5%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 56.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.5:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {x}^{2}}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.5)
   (/ 1.0 (pow (hypot 1.0 (tan x)) 2.0))
   (/ (- 1.0 (pow x 2.0)) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.5) {
		tmp = 1.0 / pow(hypot(1.0, tan(x)), 2.0);
	} else {
		tmp = (1.0 - pow(x, 2.0)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.5)
		tmp = Float64(1.0 / (hypot(1.0, tan(x)) ^ 2.0));
	else
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.5], N[(1.0 / N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $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)) < 1.5

   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-sqr-sqrt 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. pow2 99.4%

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

      4. hypot-1-def 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. *-lft-identity 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 74.7%

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

   if 1.5 < (*.f64 (tan.f64 x) (tan.f64 x))

   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 4.1%

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

   4. Taylor expanded in x around 0 9.4%

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

      1. +-commutative 9.4%

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

      2. unpow2 9.4%

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

      3. fma-define 9.4%

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

   6. Simplified 9.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.5:\\ \;\;\;\;\frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {x}^{2}}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.5:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {x}^{2}}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.5)
   (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
   (/ (- 1.0 (pow x 2.0)) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.5) {
		tmp = 1.0 / (1.0 + pow(tan(x), 2.0));
	} else {
		tmp = (1.0 - pow(x, 2.0)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.5)
		tmp = Float64(1.0 / Float64(1.0 + (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - (x ^ 2.0)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.5], N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $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)) < 1.5

   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 99.6%

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

      2. pow2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. add-log-exp 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. log-prod 99.5%

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

      4. metadata-eval 99.5%

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

      5. add-log-exp 99.6%

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

      6. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-lft-identity 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 74.7%

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

   if 1.5 < (*.f64 (tan.f64 x) (tan.f64 x))

   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 4.1%

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

   4. Taylor expanded in x around 0 9.4%

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

      1. +-commutative 9.4%

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

      2. unpow2 9.4%

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

      3. fma-define 9.4%

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

   6. Simplified 9.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.5:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {x}^{2}}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \]
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 99.5%

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

   1. div-sub 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. add-sqr-sqrt 99.3%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 54.5% 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 99.5%

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

   1. add-log-exp 99.6%

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

   2. pow2 99.6%

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

4. Applied egg-rr 99.6%

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

   1. add-log-exp 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. log-prod 99.5%

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

   4. metadata-eval 99.5%

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

   5. add-log-exp 99.6%

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

   6. pow2 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-lft-identity 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in x around 0 58.1%

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

10. Final simplification 58.1%

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

Alternative 6: 54.2% 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 99.5%

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

   1. div-sub 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. add-sqr-sqrt 99.3%

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

   4. prod-diff 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 57.8%

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

8. Final simplification 57.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))