Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.8s

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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := \cos \left(x \cdot -2\right)\\ \frac{1 - \frac{\mathsf{fma}\left(t\_0, -0.5, 0.5\right)}{\mathsf{fma}\left(0.5, t\_0, 0.5\right)}}{1 + \frac{1 - t\_1}{1 + t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (cos (* x -2.0))))
   (/
    (- 1.0 (/ (fma t_0 -0.5 0.5) (fma 0.5 t_0 0.5)))
    (+ 1.0 (/ (- 1.0 t_1) (+ 1.0 t_1))))))

C

double code(double x) {
	double t_0 = cos((x + x));
	double t_1 = cos((x * -2.0));
	return (1.0 - (fma(t_0, -0.5, 0.5) / fma(0.5, t_0, 0.5))) / (1.0 + ((1.0 - t_1) / (1.0 + t_1)));
}

Julia

function code(x)
	t_0 = cos(Float64(x + x))
	t_1 = cos(Float64(x * -2.0))
	return Float64(Float64(1.0 - Float64(fma(t_0, -0.5, 0.5) / fma(0.5, t_0, 0.5))) / Float64(1.0 + Float64(Float64(1.0 - t_1) / Float64(1.0 + t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - N[(N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] / N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + t$95$1), $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. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. lower-/.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. sqr-cos-a N/A

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

   13. lower-+.f64 N/A

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

   14. cos-2 N/A

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

   15. cos-sum N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-+.f64 99.2

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-neg-in N/A

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

   9. distribute-rgt1-in N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. lower-/.f64 N/A

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

7. Simplified 99.2%

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

   1. tan-quot N/A

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

   2. lift-cos.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-cos.f64 N/A

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

   5. frac-times N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{1 - \frac{\sin x \cdot \sin x}{{\cos x}^{\color{blue}{\left(\frac{4}{2}\right)}}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\sin x \cdot \sin x}{{\cos x}^{\left(\frac{4}{2}\right)}}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   9. sqr-sin-a N/A

      \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   10. count-2 N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(x + x\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(x + x\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   12. lift-cos.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(x + x\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

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

       \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(x + x\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{2} + \color{blue}{\frac{-1}{2}} \cdot \cos \left(x + x\right)}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{2} + \color{blue}{\cos \left(x + x\right) \cdot \frac{-1}{2}}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   16. +-commutative N/A

       \[\leadsto \frac{1 - \frac{\color{blue}{\cos \left(x + x\right) \cdot \frac{-1}{2} + \frac{1}{2}}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{1 - \frac{\color{blue}{\mathsf{fma}\left(\cos \left(x + x\right), \frac{-1}{2}, \frac{1}{2}\right)}}{{\cos x}^{\left(\frac{4}{2}\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]

   18. metadata-eval N/A

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

   19. pow2 N/A

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

   20. lift-cos.f64 N/A

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

   21. lift-cos.f64 N/A

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

   22. sqr-cos-a N/A

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

9. Applied egg-rr 99.6%

   \[\leadsto \frac{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right)}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}}}{1 + \frac{1 - \cos \left(x \cdot -2\right)}{1 + \cos \left(x \cdot -2\right)}} \]
10. Add Preprocessing

Alternative 2: 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-tan.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

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

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

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

   4. lift-neg.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. +-commutative N/A

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

6. Applied egg-rr 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.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

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. inv-pow N/A

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

   7. pow-prod-up N/A

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

   8. lower-pow.f64 N/A

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

   9. clear-num N/A

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

   10. tan-quot N/A

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

   11. lift-tan.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval 99.4

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

4. Applied egg-rr 99.4%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 99.4

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

6. Applied egg-rr 99.4%

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

   1. lift-tan.f64 N/A

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

   2. inv-pow N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. lift-pow.f64 99.5

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

8. Applied egg-rr 99.5%

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

Alternative 5: 60.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 1.5\right)}{1 - {\tan x}^{2}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (fma (cos (+ x x)) -0.5 1.5) (- 1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	return 1.0 / (fma(cos((x + x)), -0.5, 1.5) / (1.0 - pow(tan(x), 2.0)));
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(cos(Float64(x + x)), -0.5, 1.5) / Float64(1.0 - (tan(x) ^ 2.0))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * -0.5 + 1.5), $MachinePrecision] / 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. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. lower-/.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. sqr-cos-a N/A

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

   13. lower-+.f64 N/A

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

   14. cos-2 N/A

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

   15. cos-sum N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-+.f64 99.2

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

4. Applied egg-rr 99.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]
5. Step-by-step derivation

   1. tan-quot N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. frac-times N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sqr-sin-a N/A

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

   11. count-2 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. sqr-cos-a N/A

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

   19. count-2 N/A

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

   20. lift-+.f64 N/A

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

   21. lift-cos.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. lift-+.f64 N/A

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0

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

9. Simplified 64.9%

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

11. Applied egg-rr 64.9%

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 1.5\right)}{1 - {\tan x}^{2}}}} \]
12. Add Preprocessing

Alternative 6: 60.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 1.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (pow (tan x) 2.0)) (fma (cos (+ x x)) -0.5 1.5)))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / fma(cos((x + x)), -0.5, 1.5);
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / fma(cos(Float64(x + x)), -0.5, 1.5))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * -0.5 + 1.5), $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. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. lower-/.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. sqr-cos-a N/A

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

   13. lower-+.f64 N/A

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

   14. cos-2 N/A

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

   15. cos-sum N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-+.f64 99.2

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

4. Applied egg-rr 99.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]
5. Step-by-step derivation

   1. tan-quot N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. frac-times N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. sqr-sin-a N/A

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

   11. count-2 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift--.f64 N/A

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

   16. lift-cos.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. sqr-cos-a N/A

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

   19. count-2 N/A

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

   20. lift-+.f64 N/A

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

   21. lift-cos.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. lift-+.f64 N/A

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in x around 0

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

9. Simplified 64.9%

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

11. Applied egg-rr 64.9%

    \[\leadsto \color{blue}{\frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 1.5\right)}} \]
12. Add Preprocessing

Alternative 7: 54.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / 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-tan.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

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 59.9%

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

Alternative 8: 58.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 62.9%

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

6. Final simplification 62.9%

   \[\leadsto 1 - \tan x \cdot \tan x \]
7. Add Preprocessing

Alternative 9: 54.1% 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

4. Applied egg-rr 59.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))