Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.3s

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

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

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--.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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 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 99.5%

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

   1. lift--.f64 N/A

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

   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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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 99.5

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

   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. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + {\tan x}^{2}} \]
8. 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--.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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. sub-neg N/A

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

   8. lift--.f64 99.5

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{1 - {\tan x}^{2}}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
9. 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--.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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-*r/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. div-sub N/A

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

6. Applied rewrites 99.5%

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

Alternative 5: 61.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. div-inv N/A

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

   10. unpow-prod-down N/A

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

   11. pow2 N/A

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

   12. lower-fma.f64 N/A

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

6. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot x\right), {\left(\frac{1}{\cos x}\right)}^{2}, 1\right)}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. cos-2 N/A

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

   11. cos-sum N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. metadata-eval 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 58.8%

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

Alternative 6: 59.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \sin x \cdot \frac{\tan x}{\cos x}}{\color{blue}{1}} \]
6. Step-by-step derivation

7. Applied rewrites 56.3%

   \[\leadsto \frac{1 - \sin x \cdot \frac{\tan x}{\cos x}}{\color{blue}{1}} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-tan.f64 N/A

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

   11. pow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. neg-mul-1 N/A

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

   16. distribute-lft-in N/A

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

   17. metadata-eval N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. associate-/r/ N/A

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

   21. div-inv N/A

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

   22. associate-/r* N/A

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

   23. metadata-eval N/A

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

9. Applied rewrites 56.3%

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

Alternative 7: 59.8% 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

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

   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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
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-tan.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-*r/ N/A

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

   13. lift-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. div-sub N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 56.3%

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

Alternative 8: 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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))