Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 15.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $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[(t$95$0 * 0.5 + -0.5), $MachinePrecision] / N[(t$95$0 * -0.5 + -0.5), $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. /-lowering-/.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. --lowering--.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. *-lowering-*.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. cos-lowering-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. +-lowering-+.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. +-lowering-+.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. *-lowering-*.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. cos-lowering-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. +-lowering-+.f64 98.9

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

4. Applied egg-rr 98.9%

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

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

   8. /-lowering-/.f64 N/A

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

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. Step-by-step derivation

   1. frac-2neg N/A

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

8. Applied egg-rr 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{1 + {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (- 0.0 (tan x)) (tan x) 1.0) (+ 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return fma((0.0 - tan(x)), tan(x), 1.0) / (1.0 + pow(tan(x), 2.0));
}

Julia

function code(x)
	return Float64(fma(Float64(0.0 - tan(x)), tan(x), 1.0) / Float64(1.0 + (tan(x) ^ 2.0)))
end

Wolfram

code[x_] := N[(N[(N[(0.0 - N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

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

   3. pow2 N/A

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

   4. pow-lowering-pow.f64 N/A

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

   5. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. tan-lowering-tan.f64 99.5

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{1 + {\tan x}^{2}} \]
10. 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. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. tan-lowering-tan.f64 N/A

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

   4. tan-lowering-tan.f64 99.5

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

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. pow2 N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. tan-lowering-tan.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. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

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

   3. pow2 N/A

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

   4. pow-lowering-pow.f64 N/A

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

   5. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. sub0-neg N/A

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

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

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

   4. --lowering--.f64 N/A

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

   5. pow2 N/A

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

   6. pow-lowering-pow.f64 N/A

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

   7. tan-lowering-tan.f64 99.5

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 61.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x + x), $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[(0.5 - N[(t$95$0 * 0.5), $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. /-lowering-/.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. --lowering--.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. *-lowering-*.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. cos-lowering-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. +-lowering-+.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. +-lowering-+.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. *-lowering-*.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. cos-lowering-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. +-lowering-+.f64 98.9

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

4. Applied egg-rr 98.9%

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

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

   8. /-lowering-/.f64 N/A

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

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 58.8%

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

10. Final simplification 58.8%

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

Alternative 6: 61.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (fma (- 0.5 (* 0.5 (cos (* x 2.0)))) 1.0 1.0)))

C

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

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / fma(Float64(0.5 - Float64(0.5 * cos(Float64(x * 2.0)))), 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[(0.5 - N[(0.5 * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + 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. +-commutative N/A

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

   2. tan-quot N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. tan-lowering-tan.f64 N/A

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

   9. cos-lowering-cos.f64 99.4

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

4. Applied egg-rr 99.4%

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. tan-quot N/A

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

   4. pow2 N/A

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

   5. tan-quot N/A

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

   6. div-inv N/A

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

   7. unpow-prod-down N/A

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

   8. pow2 N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

6. Applied egg-rr 99.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), {\left(\frac{1}{\cos x}\right)}^{2}, 1\right)}} \]

7. Taylor expanded in x around 0

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

9. Simplified 58.8%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), \color{blue}{1}, 1\right)} \]

10. Final simplification 58.8%

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

Alternative 7: 55.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 N/A

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

   7. tan-lowering-tan.f64 N/A

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

   8. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

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

   3. pow2 N/A

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

   4. pow-lowering-pow.f64 N/A

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

   5. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 52.2%

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

10. Final simplification 52.2%

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

Alternative 8: 59.7% 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 56.4%

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

6. Final simplification 56.4%

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

Alternative 9: 55.5% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing

4. Applied egg-rr 51.8%

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

Reproduce

herbie shell --seed 2024196 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))