Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.1s

Alternatives: 6

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.6:\\ \;\;\;\;\frac{1 - {\tan x}^{2}}{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-1} - {\sin x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.6)
   (/ (- 1.0 (pow (tan x) 2.0)) 1.0)
   (- (pow (fma x x 1.0) -1.0) (pow (sin x) 2.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.6) {
		tmp = (1.0 - pow(tan(x), 2.0)) / 1.0;
	} else {
		tmp = pow(fma(x, x, 1.0), -1.0) - pow(sin(x), 2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.6)
		tmp = Float64(Float64(1.0 - (tan(x) ^ 2.0)) / 1.0);
	else
		tmp = Float64((fma(x, x, 1.0) ^ -1.0) - (sin(x) ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.6], N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[Power[N[(x * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1.6000000000000001

   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. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.5

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

   4. Applied rewrites 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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. lower--.f64 99.5

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 69.7%

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

   if 1.6000000000000001 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 99.6

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

   4. Applied rewrites 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. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. lift-*.f64 N/A

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

      8. div-sub N/A

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

      9. lower--.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. unpow2 N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. *-inverses N/A

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

      7. *-lft-identity N/A

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

      8. unpow2 N/A

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

      9. cos-sin-sum N/A

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

      10. /-rgt-identity N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 99.0

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. lower-fma.f64 24.0

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

   12. Applied rewrites 24.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{t\_0 - -1} \end{array} \end{array} \]

FPCore

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

C

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

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) / (t_0 - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (t_0 - -1.0);
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[(t$95$0 - -1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 58.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. sub-neg N/A

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

   8. lower--.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 58.2%

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

Alternative 5: 55.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / ((1.0 - math.tan(x)) / 1.0)

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(N[(1.0 - N[Tan[x], $MachinePrecision]), $MachinePrecision] / 1.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-fma.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. lift-*.f64 N/A

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

   8. difference-of-sqr--1 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower--.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 55.8%

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

11. Applied rewrites 54.6%

    \[\leadsto \color{blue}{\frac{1}{\frac{1 - \tan x}{1}}} \]
12. Add Preprocessing

Alternative 6: 54.5% accurate, 428.0× 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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 54.0%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))