Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 14.0s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot N/A

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

   2. div-inv N/A

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

   3. tan-quot N/A

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

   4. div-inv N/A

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

   5. swap-sqr N/A

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

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

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

   7. sqr-sin-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. inv-pow N/A

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

   15. inv-pow N/A

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

   16. pow-prod-down N/A

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

   17. inv-pow N/A

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

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

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

   19. sqr-cos-a N/A

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

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

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

   21. cos-2 N/A

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

   22. cos-sum N/A

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

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

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

4. Applied egg-rr 99.1%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

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

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

6. Applied egg-rr 99.5%

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

   1. un-div-inv N/A

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

   2. *-rgt-identity N/A

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

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

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

   4. *-rgt-identity N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

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

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

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

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

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

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

   12. +-commutative N/A

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

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

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

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

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

   15. +-lowering-+.f64 99.6

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

8. Applied egg-rr 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

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

   2. pow2 N/A

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

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

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

   4. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

   1. pow2 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

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

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

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

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

   7. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 3: 61.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(t_0 * 0.5)
	return Float64(Float64(1.0 + Float64(Float64(1.0 / Float64(0.5 + t_1)) * Float64(t_1 - 0.5))) / Float64(1.0 + fma(t_0, -0.5, 0.5)))
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[(1.0 / N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * -0.5 + 0.5), $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

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

   2. div-inv N/A

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

   3. tan-quot N/A

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

   4. div-inv N/A

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

   5. swap-sqr N/A

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

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

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

   7. sqr-sin-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. inv-pow N/A

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

   15. inv-pow N/A

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

   16. pow-prod-down N/A

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

   17. inv-pow N/A

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

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

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

   19. sqr-cos-a N/A

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

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

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

   21. cos-2 N/A

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

   22. cos-sum N/A

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

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

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

4. Applied egg-rr 99.1%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. count-2 N/A

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

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

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

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

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 61.4%

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

10. Final simplification 61.4%

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

Alternative 4: 59.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 59.6%

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

8. Final simplification 59.6%

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

Alternative 5: 59.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. pow2 N/A

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

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

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

   4. tan-lowering-tan.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 59.6%

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

8. Final simplification 59.6%

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

Alternative 6: 55.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. div-inv N/A

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

   3. tan-quot N/A

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

   4. div-inv N/A

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

   5. swap-sqr N/A

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

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

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

   7. sqr-sin-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. inv-pow N/A

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

   15. inv-pow N/A

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

   16. pow-prod-down N/A

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

   17. inv-pow N/A

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

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

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

   19. sqr-cos-a N/A

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

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

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

   21. cos-2 N/A

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

   22. cos-sum N/A

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

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

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

4. Applied egg-rr 99.1%

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

5. Taylor expanded in x around 0

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

7. Simplified 57.3%

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

8. Taylor expanded in x around 0

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

10. Simplified 55.8%

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

11. Final simplification 55.8%

    \[\leadsto 1 + \left(\cos \left(x + x\right) \cdot 0.5 - 0.5\right) \]
12. Add Preprocessing

Alternative 7: 55.1% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

4. Applied egg-rr 55.4%

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

Reproduce

herbie shell --seed 2024197 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))