Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 7.4s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $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-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t\_0 \leq 0.6:\\ \;\;\;\;\frac{1 - 1 \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right)}{t\_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - {\tan x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= t_0 0.6)
     (/ (- 1.0 (* 1.0 (- 0.5 (* (cos (+ x x)) 0.5)))) (+ t_0 1.0))
     (* 1.0 (- 1.0 (pow (tan x) 2.0))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (1.0 - (1.0 * (0.5 - (cos((x + x)) * 0.5)))) / (t_0 + 1.0);
	} else {
		tmp = 1.0 * (1.0 - pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    if (t_0 <= 0.6d0) then
        tmp = (1.0d0 - (1.0d0 * (0.5d0 - (cos((x + x)) * 0.5d0)))) / (t_0 + 1.0d0)
    else
        tmp = 1.0d0 * (1.0d0 - (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (1.0 - (1.0 * (0.5 - (Math.cos((x + x)) * 0.5)))) / (t_0 + 1.0);
	} else {
		tmp = 1.0 * (1.0 - Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	tmp = 0
	if t_0 <= 0.6:
		tmp = (1.0 - (1.0 * (0.5 - (math.cos((x + x)) * 0.5)))) / (t_0 + 1.0)
	else:
		tmp = 1.0 * (1.0 - math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (t_0 <= 0.6)
		tmp = Float64(Float64(1.0 - Float64(1.0 * Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)))) / Float64(t_0 + 1.0));
	else
		tmp = Float64(1.0 * Float64(1.0 - (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	tmp = 0.0;
	if (t_0 <= 0.6)
		tmp = (1.0 - (1.0 * (0.5 - (cos((x + x)) * 0.5)))) / (t_0 + 1.0);
	else
		tmp = 1.0 * (1.0 - (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6], N[(N[(1.0 - N[(1.0 * N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 0.599999999999999978

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. div-inv N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. div-inv N/A

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

      8. swap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. sqr-sin-a N/A

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

      11. lower--.f64 N/A

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

      12. cos-2 N/A

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

      13. cos-sum N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. inv-pow N/A

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

      18. inv-pow N/A

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

      19. pow-prod-down N/A

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

      20. inv-pow N/A

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

      21. lower-/.f64 N/A

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

      22. sqr-cos-a N/A

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

      23. lower-+.f64 N/A

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

      24. cos-2 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \frac{1 - \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}}{1 + \tan x \cdot \tan x} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 76.9%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \color{blue}{1}}{1 + \tan x \cdot \tan x} \]

   if 0.599999999999999978 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.0%

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

      1. 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 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lift-+.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-pow.f64 N/A

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

      18. lift--.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 16.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;\frac{1 - 1 \cdot \left(0.5 - \cos \left(x + x\right) \cdot 0.5\right)}{\tan x \cdot \tan x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 - {\tan x}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 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

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 99.5

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

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 5: 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-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 99.5

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 6: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{\frac{0.5 - \cos \left(x + x\right) \cdot 0.5}{1} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ (/ (- 0.5 (* (cos (+ x x)) 0.5)) 1.0) 1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (((0.5 - (cos((x + x)) * 0.5)) / 1.0) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) * tan(x))) / (((0.5d0 - (cos((x + x)) * 0.5d0)) / 1.0d0) + 1.0d0)
end function

Java

public static double code(double x) {
	return (1.0 - (Math.tan(x) * Math.tan(x))) / (((0.5 - (Math.cos((x + x)) * 0.5)) / 1.0) + 1.0);
}

Python

def code(x):
	return (1.0 - (math.tan(x) * math.tan(x))) / (((0.5 - (math.cos((x + x)) * 0.5)) / 1.0) + 1.0)

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(Float64(Float64(0.5 - Float64(cos(Float64(x + x)) * 0.5)) / 1.0) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) * tan(x))) / (((0.5 - (cos((x + x)) * 0.5)) / 1.0) + 1.0);
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / 1.0), $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

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

   8. sqr-sin-a N/A

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

   9. lower--.f64 N/A

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

   10. cos-2 N/A

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

   11. cos-sum N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. sqr-cos-a N/A

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

   16. lower-+.f64 N/A

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

   17. cos-2 N/A

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

   18. cos-sum N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cos.f64 N/A

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

   21. lower-+.f64 99.0

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \color{blue}{\left(x + x\right)}}} \]

4. Applied rewrites 99.0%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 62.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{\color{blue}{1}}} \]

8. Final simplification 62.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{0.5 - \cos \left(x + x\right) \cdot 0.5}{1} + 1} \]
9. Add Preprocessing

Alternative 7: 59.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / (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 / (1.0d0 - (tan(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 / N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $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. 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. Taylor expanded in x around 0

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

7. Applied rewrites 59.9%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lift-neg.f64 N/A

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

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

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

   9. pow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower-/.f64 59.9

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

9. Applied rewrites 59.9%

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

Alternative 8: 59.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \left(1 - {\tan x}^{2}\right) \end{array} \]

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. 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

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

   2. frac-2neg N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. lift-+.f64 N/A

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

   12. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-pow.f64 N/A

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

   18. lift--.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 59.9%

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

Alternative 9: 55.3% 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 56.2%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))