Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{{\tan x}^{2} - -1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (- 0.0 (tan x)) (tan x) 1.0) (- (pow (tan x) 2.0) -1.0)))

C

double code(double x) {
	return fma((0.0 - tan(x)), tan(x), 1.0) / (pow(tan(x), 2.0) - -1.0);
}

Julia

function code(x)
	return Float64(fma(Float64(0.0 - tan(x)), tan(x), 1.0) / Float64((tan(x) ^ 2.0) - -1.0))
end

Wolfram

code[x_] := N[(N[(N[(0.0 - N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right) + 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x + 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\tan x\right), \tan x, 1\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\tan x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(0 - \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left(1 + {\tan x}^{\color{blue}{2}}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} + \color{blue}{1}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} - \color{blue}{-1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\left({\tan x}^{2}\right), \color{blue}{-1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\tan x, 2\right), -1\right)\right) \]

   7. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{\color{blue}{{\tan x}^{2} - -1}} \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((-1.0 - tan(x)) * (tan(x) + -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)) * (tan(x) + (-1.0d0))) / ((tan(x) ** 2.0d0) - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right) + 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x + 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\tan x\right), \tan x, 1\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\tan x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(0 - \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left(1 + {\tan x}^{\color{blue}{2}}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} + \color{blue}{1}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \left({\tan x}^{2} - \color{blue}{-1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\left({\tan x}^{2}\right), \color{blue}{-1}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\tan x, 2\right), -1\right)\right) \]

   7. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{\color{blue}{{\tan x}^{2} - -1}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(0 - \tan x\right) \cdot \tan x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

   2. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \tan x \cdot \tan x\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - {\tan x}^{2}\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{2}\right), -1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{2}\right), -1\right)\right) \]

   7. tan-lowering-tan.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

8. Applied egg-rr 99.4%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left({\tan x}^{2}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left({\tan x}^{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\mathsf{tan.f64}\left(x\right)}, 2\right), -1\right)\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(-1 + {\tan x}^{2}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left({\tan x}^{2} + -1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\mathsf{tan.f64}\left(x\right)}, 2\right), -1\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - \left({\tan x}^{2} + -1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - \left(\tan x \cdot \tan x + -1\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

   7. difference-of-sqr--1 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(0 - \left(\tan x + 1\right) \cdot \left(\tan x - 1\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{2}\right), -1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(0 + \left(\mathsf{neg}\left(\left(\tan x + 1\right)\right)\right) \cdot \left(\tan x - 1\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \left(\left(\mathsf{neg}\left(\left(\tan x + 1\right)\right)\right) \cdot \left(\tan x - 1\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)}, -1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(\tan x + 1\right)\right)\right), \left(\tan x - 1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), \color{blue}{2}\right), -1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\left(\tan x + 1\right)\right), \left(\tan x - 1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\tan x, 1\right)\right), \left(\tan x - 1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), 1\right)\right), \left(\tan x - 1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), 1\right)\right), \left(\tan x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), 1\right)\right), \left(\tan x + -1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), 1\right)\right), \mathsf{+.f64}\left(\tan x, -1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

   17. tan-lowering-tan.f64 99.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{neg.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), 1\right)\right), \mathsf{+.f64}\left(\mathsf{tan.f64}\left(x\right), -1\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right) \]

10. Applied egg-rr 99.4%

    \[\leadsto \frac{\color{blue}{0 + \left(-\left(\tan x + 1\right)\right) \cdot \left(\tan x + -1\right)}}{{\tan x}^{2} - -1} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\left(-1 - \tan x\right) \cdot \left(\tan x + -1\right)}{{\tan x}^{2} - -1} \]
12. 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 99.4%

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

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

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

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\tan x \cdot \tan x\right)\right), \left(\color{blue}{1} + \tan x \cdot \tan x\right)\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\tan x}^{2}\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \left(1 + \tan x \cdot \tan x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\tan x \cdot \tan x\right)}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \left({\tan x}^{\color{blue}{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right) \]

4. Applied egg-rr 99.4%

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

Alternative 4: 59.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

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

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

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

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

   7. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\tan x}^{2}\right)\right)\right), \left(\frac{1}{1 - \tan x \cdot \tan x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\tan x, 2\right)\right)\right), \left(\frac{1}{1 - \tan x \cdot \tan x}\right)\right) \]

   9. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \left(\frac{1}{1 - \tan x \cdot \tan x}\right)\right) \]

   10. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(1 - \tan x \cdot \tan x\right)\right)}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\left(1 - \tan x \cdot \tan x\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\left(1 - \tan x \cdot \tan x\right)\right)\right)}\right)\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(0 - \color{blue}{\left(1 - \tan x \cdot \tan x\right)}\right)\right)\right) \]

   14. associate--r- N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(\left(0 - 1\right) + \color{blue}{\tan x \cdot \tan x}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(-1 + \color{blue}{\tan x} \cdot \tan x\right)\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \left(\tan x \cdot \tan x + \color{blue}{-1}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\left(\tan x \cdot \tan x\right), \color{blue}{-1}\right)\right)\right) \]

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right), -1\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 58.3%

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

Alternative 5: 59.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x \cdot \tan x\right)\right) + 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\tan x\right)\right) \cdot \tan x + 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\tan x\right), \tan x, 1\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\tan x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(0 - \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \tan x\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \tan x, 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(x\right), \mathsf{tan.f64}\left(x\right)\right)\right)\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}}{1 + \tan x \cdot \tan x} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{tan.f64}\left(x\right)\right), \mathsf{tan.f64}\left(x\right), 1\right), \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 58.3%

   \[\leadsto \frac{\mathsf{fma}\left(0 - \tan x, \tan x, 1\right)}{\color{blue}{1}} \]
8. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \left(0 - \tan x\right) \cdot \tan x + \color{blue}{1} \]

   2. +-commutative N/A

      \[\leadsto 1 + \color{blue}{\left(0 - \tan x\right) \cdot \tan x} \]

   3. sub0-neg N/A

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

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

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

   5. unpow2 N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left({\tan x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\tan x, \color{blue}{2}\right)\right) \]

   8. tan-lowering-tan.f64 58.3%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{tan.f64}\left(x\right), 2\right)\right) \]

9. Applied egg-rr 58.3%

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

Alternative 6: 55.2% accurate, 411.0× 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 99.4%

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

3. Taylor expanded in x around 0

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

5. Simplified 54.0%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))