Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

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

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. lower-/.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. sqr-cos-a N/A

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

   13. lower-+.f64 N/A

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

   14. cos-2 N/A

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

   15. cos-sum N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(x + x\right)}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}} \]
5. 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. lift-+.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sqr-cos-a N/A

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

   11. count-2 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lift-/.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. un-div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. unsub-neg N/A

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

   10. distribute-lft-neg-out N/A

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

   11. lift-neg.f64 N/A

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

   12. +-commutative N/A

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

6. Applied rewrites 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 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. un-div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. unsub-neg N/A

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

   10. distribute-lft-neg-out N/A

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

   11. lift-neg.f64 N/A

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

   12. +-commutative N/A

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

6. Applied rewrites 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. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. pow2 N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. lift-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. lift--.f64 N/A

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

8. Applied rewrites 99.5%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[Sin[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. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. pow-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. div-sub N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. unpow2 N/A

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

   4. associate-*r/ N/A

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

   5. associate-*l/ N/A

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

   6. *-inverses N/A

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

   7. *-lft-identity N/A

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

   8. unpow2 N/A

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

   9. cos-sin-sum N/A

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

   10. /-rgt-identity N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 61.6% 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. lower-/.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. sqr-cos-a N/A

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

   13. lower-+.f64 N/A

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

   14. cos-2 N/A

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

   15. cos-sum N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 63.2%

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

8. Final simplification 63.2%

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

Alternative 7: 55.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lift-tan.f64 N/A

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

   9. tan-quot N/A

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

   10. associate-*r/ N/A

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

   11. div-inv N/A

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

   12. lower-fma.f64 N/A

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

   13. distribute-lft-neg-out N/A

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

   14. *-commutative N/A

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

   15. lower-neg.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. lift-pow.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 57.6%

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

Alternative 8: 59.7% 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. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.5%

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

10. Final simplification 61.5%

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

Alternative 9: 55.5% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 57.3%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))