Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.6s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 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 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. div-sub N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-to-exp N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-exp.f64 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lower-fma.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

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 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. div-sub N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-to-exp N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-exp.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 4: 60.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. lower-/.f64 N/A

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

   15. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 61.2

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

9. Applied rewrites 61.2%

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

Alternative 5: 59.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. div-sub N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-to-exp N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-exp.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.0%

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

    1. lift-pow.f64 N/A

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

    2. pow2 N/A

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

    3. lift-tan.f64 N/A

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

    4. lift-tan.f64 N/A

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

    5. tan-quot N/A

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

    6. lift-sin.f64 N/A

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

    7. lift-cos.f64 N/A

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

    8. tan-quot N/A

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

    9. lift-sin.f64 N/A

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

    10. lift-cos.f64 N/A

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

    11. frac-times N/A

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

    12. lift-sin.f64 N/A

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

    13. lift-sin.f64 N/A

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

    14. sqr-sin-a N/A

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

    15. lift-*.f64 N/A

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

    16. lift-cos.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. lift--.f64 N/A

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

    19. lift-cos.f64 N/A

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

    20. lift-cos.f64 N/A

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

    21. sqr-cos-a N/A

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

    22. lift-*.f64 N/A

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

    23. lift-cos.f64 N/A

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

    24. lift-*.f64 N/A

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

    25. lift-+.f64 N/A

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

11. Applied rewrites 61.0%

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

Alternative 6: 59.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. div-sub N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-to-exp N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-exp.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.0%

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

    1. lift-/.f64 N/A

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

    2. lift--.f64 N/A

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

    3. div-sub N/A

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

    4. frac-sub N/A

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

    5. clear-num N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 N/A

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

    8. lower-*.f64 N/A

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

11. Applied rewrites 61.0%

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

12. Final simplification 61.0%

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

Alternative 7: 59.9% 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 99.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. div-sub N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow-to-exp N/A

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

   12. lift-log.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-exp.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.0%

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

Alternative 8: 55.7% 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 99.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 56.5%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))