Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.2s

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, 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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. unpow1 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 59.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{t\_0 + 1} \leq -0.02:\\ \;\;\;\;-1.6666666666666667\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ t_0 1.0)) -0.02) -1.6666666666666667 1.0)))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (t_0 + 1.0)) <= -0.02) {
		tmp = -1.6666666666666667;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    if (((1.0d0 - t_0) / (t_0 + 1.0d0)) <= (-0.02d0)) then
        tmp = -1.6666666666666667d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double tmp;
	if (((1.0 - t_0) / (t_0 + 1.0)) <= -0.02) {
		tmp = -1.6666666666666667;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	tmp = 0
	if ((1.0 - t_0) / (t_0 + 1.0)) <= -0.02:
		tmp = -1.6666666666666667
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(t_0 + 1.0)) <= -0.02)
		tmp = -1.6666666666666667;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	tmp = 0.0;
	if (((1.0 - t_0) / (t_0 + 1.0)) <= -0.02)
		tmp = -1.6666666666666667;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], -0.02], -1.6666666666666667, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < -0.0200000000000000004

   1. Initial program 99.3%

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

      1. rem-exp-log N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. log-pow N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-log.f64 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}} + \frac{1}{1 + {x}^{2}}\right)\right) + \frac{1}{1 + {x}^{2}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}} + \frac{1}{1 + {x}^{2}}\right)\right)\right)} + \frac{1}{1 + {x}^{2}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}\right) \cdot {x}^{2} + \frac{1}{1 + {x}^{2}} \cdot {x}^{2}\right)}\right)\right) + \frac{1}{1 + {x}^{2}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}\right) \cdot {x}^{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{1 + {x}^{2}} \cdot {x}^{2}\right)\right)\right)} + \frac{1}{1 + {x}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}\right) \cdot {x}^{2}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \frac{1}{1 + {x}^{2}}}\right)\right)\right) + \frac{1}{1 + {x}^{2}} \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{2}{3} \cdot \frac{{x}^{2}}{{\left(1 + {x}^{2}\right)}^{2}}\right) \cdot {x}^{2}\right)\right) + \left(\left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{1 + {x}^{2}}\right)\right) + \frac{1}{1 + {x}^{2}}\right)} \]

   7. Applied rewrites 3.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{x}^{4}}{{\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{2}}, -0.6666666666666666, \left(1 - x \cdot x\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-5}{3} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.1%

       \[\leadsto -1.6666666666666667 \]

   if -0.0200000000000000004 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1} \leq -0.02:\\ \;\;\;\;-1.6666666666666667\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. unpow1 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (* (tan x) (tan x)))
  (+
   (pow
    (/
     (fma
      (fma
       (fma -0.0021164021164021165 (* x x) -0.022222222222222223)
       (* x x)
       -0.3333333333333333)
      (* x x)
      1.0)
     x)
    -2.0)
   1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (pow((fma(fma(fma(-0.0021164021164021165, (x * x), -0.022222222222222223), (x * x), -0.3333333333333333), (x * x), 1.0) / x), -2.0) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64((Float64(fma(fma(fma(-0.0021164021164021165, Float64(x * x), -0.022222222222222223), Float64(x * x), -0.3333333333333333), Float64(x * x), 1.0) / x) ^ -2.0) + 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(N[(N[(-0.0021164021164021165 * N[(x * x), $MachinePrecision] + -0.022222222222222223), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], -2.0], $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-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) \cdot {x}^{2}} + 1}{x}\right)}^{-2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, {x}^{2}, 1\right)}}{x}\right)}^{-2}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, {x}^{2}, \frac{-1}{3}\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   9. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945} \cdot {x}^{2} + \color{blue}{\frac{-1}{45}}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945}, {x}^{2}, \frac{-1}{45}\right)}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{x \cdot x}, \frac{-1}{45}\right), {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{x \cdot x}, \frac{-1}{45}\right), {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   14. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   16. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), x \cdot x, \frac{-1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

   17. lower-*.f64 59.3

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 59.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}}^{-2}} \]

8. Final simplification 59.3%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \]
9. Add Preprocessing

Alternative 5: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (* (tan x) (tan x)))
  (+
   (pow
    (/
     (fma (fma -0.022222222222222223 (* x x) -0.3333333333333333) (* x x) 1.0)
     x)
    -2.0)
   1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (pow((fma(fma(-0.022222222222222223, (x * x), -0.3333333333333333), (x * x), 1.0) / x), -2.0) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64((Float64(fma(fma(-0.022222222222222223, Float64(x * x), -0.3333333333333333), Float64(x * x), 1.0) / x) ^ -2.0) + 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(N[(-0.022222222222222223 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], -2.0], $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-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right) \cdot {x}^{2}} + 1}{x}\right)}^{-2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}, {x}^{2}, 1\right)}}{x}\right)}^{-2}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {x}^{2}, \frac{-1}{3}\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   10. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, x \cdot x, \frac{-1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

   11. lower-*.f64 59.2

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 59.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}}^{-2}} \]

8. Final simplification 59.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \]
9. Add Preprocessing

Alternative 6: 59.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (* (tan x) (tan x)))
  (+ (pow (/ (fma -0.3333333333333333 (* x x) 1.0) x) -2.0) 1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (pow((fma(-0.3333333333333333, (x * x), 1.0) / x), -2.0) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64((Float64(fma(-0.3333333333333333, Float64(x * x), 1.0) / x) ^ -2.0) + 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], -2.0], $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-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {x}^{2}}{x}\right)}}^{-2}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {x}^{2}}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\frac{-1}{3} \cdot {x}^{2} + 1}}{x}\right)}^{-2}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {x}^{2}, 1\right)}}{x}\right)}^{-2}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

   5. lower-*.f64 59.2

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

7. Applied rewrites 59.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)}{x}\right)}}^{-2}} \]

8. Final simplification 59.2%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \]
9. Add Preprocessing

Alternative 7: 59.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. unpow1 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 58.6%

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

Alternative 8: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{{\tan x}^{4} - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (- (pow (tan x) 4.0) 1.0)))

C

double code(double x) {
	return -1.0 / (pow(tan(x), 4.0) - 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / ((tan(x) ** 4.0d0) - 1.0d0)
end function

Java

public static double code(double x) {
	return -1.0 / (Math.pow(Math.tan(x), 4.0) - 1.0);
}

Python

def code(x):
	return -1.0 / (math.pow(math.tan(x), 4.0) - 1.0)

Julia

function code(x)
	return Float64(-1.0 / Float64((tan(x) ^ 4.0) - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / ((tan(x) ^ 4.0) - 1.0);
end

Wolfram

code[x_] := N[(-1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 4.0], $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-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-+ N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow-prod-down N/A

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

   8. pow-sqr N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{{\tan x}^{\color{blue}{4}} - 1 \cdot 1}{\tan x \cdot \tan x - 1}} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{\color{blue}{{\tan x}^{4}} - 1 \cdot 1}{\tan x \cdot \tan x - 1}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{{\tan x}^{4} - \color{blue}{1}}{\tan x \cdot \tan x - 1}} \]

   12. sub-neg N/A

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

   13. lift-*.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{{\tan x}^{4} - 1}{\tan x \cdot \tan x + \color{blue}{-1}}} \]

   15. lift-fma.f64 N/A

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

   16. lift--.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 57.7%

   \[\leadsto \frac{\color{blue}{-1}}{{\tan x}^{4} - 1} \]
10. Add Preprocessing

Alternative 9: 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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.2%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))