Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.4s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(t_0 * 0.5)
	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(0.5 - t_1) / Float64(0.5 + t_1))))
end

Wolfram

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

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

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

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

   14. cos-lowering-cos.f64 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

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. Final simplification 99.6%

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

Alternative 2: 60.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(t\_1, -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, t\_1, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - t\_1 \cdot -0.5\right) - 0.5}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (cos (+ x x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) 0.23)
     (- 1.0 (* (fma t_1 -0.5 0.5) (/ 1.0 (fma 0.5 t_1 0.5))))
     (/ (- (- 1.0 (* t_1 -0.5)) 0.5) (fma (tan x) (tan x) 1.0)))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = cos((x + x));
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.23) {
		tmp = 1.0 - (fma(t_1, -0.5, 0.5) * (1.0 / fma(0.5, t_1, 0.5)));
	} else {
		tmp = ((1.0 - (t_1 * -0.5)) - 0.5) / fma(tan(x), tan(x), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= 0.23)
		tmp = Float64(1.0 - Float64(fma(t_1, -0.5, 0.5) * Float64(1.0 / fma(0.5, t_1, 0.5))));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(t_1 * -0.5)) - 0.5) / fma(tan(x), tan(x), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.23], N[(1.0 - N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] * N[(1.0 / N[(0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < 0.23000000000000001

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 16.7%

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

   if 0.23000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.7%

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

      1. +-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 99.7

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

   4. Applied egg-rr 99.7%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 73.9%

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

       1. *-rgt-identity N/A

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

       2. associate--r+ N/A

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

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

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

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

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

       5. *-lowering-*.f64 N/A

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

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

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

       7. +-lowering-+.f64 73.9

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

   11. Applied egg-rr 73.9%

       \[\leadsto \frac{\color{blue}{\left(1 - \cos \left(x + x\right) \cdot -0.5\right) - 0.5}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \cos \left(x + x\right) \cdot -0.5\right) - 0.5}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(t\_1, -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, t\_1, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}} \cdot \left(0.5 - t\_1 \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (cos (+ x x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) 0.23)
     (- 1.0 (* (fma t_1 -0.5 0.5) (/ 1.0 (fma 0.5 t_1 0.5))))
     (* (/ 1.0 (+ 1.0 (pow (tan x) 2.0))) (- 0.5 (* t_1 -0.5))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = cos((x + x));
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.23) {
		tmp = 1.0 - (fma(t_1, -0.5, 0.5) * (1.0 / fma(0.5, t_1, 0.5)));
	} else {
		tmp = (1.0 / (1.0 + pow(tan(x), 2.0))) * (0.5 - (t_1 * -0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= 0.23)
		tmp = Float64(1.0 - Float64(fma(t_1, -0.5, 0.5) * Float64(1.0 / fma(0.5, t_1, 0.5))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + (tan(x) ^ 2.0))) * Float64(0.5 - Float64(t_1 * -0.5)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.23], N[(1.0 - N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] * N[(1.0 / N[(0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < 0.23000000000000001

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 16.7%

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

   if 0.23000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.7%

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

      1. +-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 99.7

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

   4. Applied egg-rr 99.7%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 73.9%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. +-commutative N/A

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

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

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

       7. pow2 N/A

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

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

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

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

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

       10. *-rgt-identity N/A

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

       11. +-commutative N/A

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

       12. associate--r+ N/A

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

       13. metadata-eval N/A

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

       14. --lowering--.f64 N/A

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

       15. *-lowering-*.f64 N/A

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

       16. cos-lowering-cos.f64 N/A

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

       17. +-lowering-+.f64 73.9

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

   11. Applied egg-rr 73.9%

       \[\leadsto \color{blue}{\frac{1}{1 + {\tan x}^{2}} \cdot \left(0.5 - \cos \left(x + x\right) \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}} \cdot \left(0.5 - \cos \left(x + x\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(t\_1, -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, t\_1, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_1 \cdot -0.5}{1 + {\tan x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))) (t_1 (cos (+ x x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) 0.23)
     (- 1.0 (* (fma t_1 -0.5 0.5) (/ 1.0 (fma 0.5 t_1 0.5))))
     (/ (- 0.5 (* t_1 -0.5)) (+ 1.0 (pow (tan x) 2.0))))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double t_1 = cos((x + x));
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= 0.23) {
		tmp = 1.0 - (fma(t_1, -0.5, 0.5) * (1.0 / fma(0.5, t_1, 0.5)));
	} else {
		tmp = (0.5 - (t_1 * -0.5)) / (1.0 + pow(tan(x), 2.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= 0.23)
		tmp = Float64(1.0 - Float64(fma(t_1, -0.5, 0.5) * Float64(1.0 / fma(0.5, t_1, 0.5))));
	else
		tmp = Float64(Float64(0.5 - Float64(t_1 * -0.5)) / Float64(1.0 + (tan(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.23], N[(1.0 - N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] * N[(1.0 / N[(0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < 0.23000000000000001

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 16.7%

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

   if 0.23000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.7%

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

      1. +-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 99.7

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

   4. Applied egg-rr 99.7%

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

      1. tan-quot N/A

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

      2. tan-quot N/A

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

      3. frac-times N/A

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

      4. sqr-sin-a N/A

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

      5. count-2 N/A

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

      6. sqr-cos-a N/A

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

      7. count-2 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

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

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

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

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

      15. cos-lowering-cos.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 73.9%

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

       1. /-lowering-/.f64 N/A

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

       2. *-rgt-identity N/A

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

       3. +-commutative N/A

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

       4. associate--r+ N/A

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

       5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

       10. +-commutative N/A

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

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

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

       12. pow2 N/A

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

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

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

       14. tan-lowering-tan.f64 73.9

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

   11. Applied egg-rr 73.9%

       \[\leadsto \color{blue}{\frac{0.5 - \cos \left(x + x\right) \cdot -0.5}{1 + {\tan x}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \leq 0.23:\\ \;\;\;\;1 - \mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \cos \left(x + x\right) \cdot -0.5}{1 + {\tan x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. +-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. cancel-sign-sub-inv N/A

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

   2. tan-quot N/A

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

   3. div-inv N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. div-inv N/A

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

   10. tan-quot N/A

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

   11. *-commutative N/A

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

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

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

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

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

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

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

   15. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

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

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

   2. pow2 N/A

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

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

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

   4. 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%

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

Alternative 7: 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. +-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. /-lowering-/.f64 N/A

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

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

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

   3. pow2 N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. pow2 N/A

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

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

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

   10. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 8: 60.5% 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

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

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

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

   14. cos-lowering-cos.f64 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

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 57.8%

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

10. Final simplification 57.8%

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

Alternative 9: 60.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.5 - N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * 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. Taylor expanded in x around 0

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

7. Simplified 57.8%

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

8. Final simplification 57.8%

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

Alternative 10: 58.6% accurate, 1.8× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

   8. div-inv N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

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

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

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

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

   15. cos-lowering-cos.f64 N/A

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

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

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

   17. metadata-eval N/A

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

   18. /-lowering-/.f64 N/A

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 55.4%

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

10. Final simplification 55.4%

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

Alternative 11: 58.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-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. clear-num N/A

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

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

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

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

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

   4. +-commutative N/A

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

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

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

   6. pow2 N/A

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

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

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

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

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

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

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

   10. pow2 N/A

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

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

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

   12. tan-lowering-tan.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in x around 0

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

9. Simplified 55.4%

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

Alternative 12: 58.6% 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 55.4%

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

6. Final simplification 55.4%

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

Alternative 13: 54.7% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- 1.0 (fma (cos (+ x x)) -0.5 0.5)))

C

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

Julia

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

Wolfram

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

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

   8. div-inv N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

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

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

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

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

   15. cos-lowering-cos.f64 N/A

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

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

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

   17. metadata-eval N/A

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

   18. /-lowering-/.f64 N/A

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 53.4%

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

10. Taylor expanded in x around 0

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

12. Simplified 51.3%

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

13. Final simplification 51.3%

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

Alternative 14: 54.3% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 50.9%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))