Trigonometry B

Percentage Accurate: 99.5% → 99.4%

Time: 10.2s

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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot 99.5%

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

   2. associate-*r/ 99.5%

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

3. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

   \[\leadsto \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \color{blue}{{\tan x}^{2}}} \]
8. Step-by-step derivation

   1. pow2 99.5%

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

   2. tan-quot 99.5%

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

   3. clear-num 99.5%

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

   4. div-inv 99.5%

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

   5. clear-num 99.5%

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

   6. tan-quot 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{\tan x}{\frac{1}{\tan x}}} \]

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. tan-quot 99.5%

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

   2. associate-*r/ 99.5%

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

3. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 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 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. 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-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 58.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.0)
   (/ 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)) <= 1.0) {
		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)) <= 1.0)
		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], 1.0], 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

   1. Initial program 99.6%

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

      1. tan-quot 99.7%

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

      2. associate-*r/ 99.7%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
   4. Step-by-step derivation

      1. add-log-exp 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. log-prod 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-log-exp 99.7%

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

      6. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-lft-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 77.7%

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

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 4.2%

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

   3. Taylor expanded in x around 0 10.3%

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

      1. +-commutative 10.3%

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

      2. unpow2 10.3%

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

      3. fma-def 10.3%

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

   5. Simplified 10.3%

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

4. Final simplification 60.6%

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

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 99.5%

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

   1. tan-quot 99.5%

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

   2. associate-*r/ 99.5%

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

3. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. hypot-1-def 99.4%

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

   3. hypot-1-def 99.4%

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

   4. unpow2 99.4%

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

   5. frac-2neg 99.4%

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

   6. div-inv 99.3%

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

5. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-mul-1 99.5%

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

   4. associate-/l* 99.4%

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

   5. metadata-eval 99.4%

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

   6. distribute-neg-in 99.4%

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

   7. neg-mul-1 99.4%

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

   8. associate-*r/ 99.4%

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

   9. associate-/r* 99.4%

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

   10. metadata-eval 99.4%

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

   11. associate-/l* 99.5%

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

   12. *-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 56.5% 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. Step-by-step derivation

   1. tan-quot 99.5%

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

   2. associate-*r/ 99.5%

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

3. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. add-log-exp 98.3%

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

   2. *-un-lft-identity 98.3%

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

   3. log-prod 98.3%

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

   4. metadata-eval 98.3%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 58.4%

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

9. Final simplification 58.4%

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

Alternative 7: 56.1% accurate, 411.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

   1. tan-quot 99.5%

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

   2. associate-*r/ 99.5%

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

3. Applied egg-rr 99.5%

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

4. Taylor expanded in x around 0 58.1%

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

5. Final simplification 58.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))