Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

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. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

   7. tan-lowering-tan.f64 99.6

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

4. Applied egg-rr 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. +-commutative N/A

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

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

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

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

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

   4. tan-lowering-tan.f64 99.6

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

6. Applied egg-rr 99.6%

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

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

   7. tan-lowering-tan.f64 99.6

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

4. Applied egg-rr 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. +-commutative N/A

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

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

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

   3. pow2 N/A

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

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

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

   5. tan-lowering-tan.f64 99.6

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

6. Applied egg-rr 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 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. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

   7. tan-lowering-tan.f64 99.6

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

4. Applied egg-rr 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. +-commutative N/A

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

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

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

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

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

   4. tan-lowering-tan.f64 99.6

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

6. Applied egg-rr 99.6%

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

   1. +-commutative N/A

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

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

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

   3. pow2 N/A

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

   4. sub-neg N/A

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

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

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

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

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

   7. tan-lowering-tan.f64 99.5

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

8. Applied egg-rr 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. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

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

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

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

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

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

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

   7. tan-lowering-tan.f64 99.6

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

4. Applied egg-rr 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. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. pow2 N/A

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

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

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

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

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

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

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

   10. pow2 N/A

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

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

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

   12. tan-lowering-tan.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 5: 59.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot -2\right)\\ 1 + \frac{1 - t\_0}{-1 - t\_0} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = cos((x * -2.0));
	return 1.0 + ((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 = cos((x * (-2.0d0)))
    code = 1.0d0 + ((1.0d0 - t_0) / ((-1.0d0) - t_0))
end function

Java

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

Python

def code(x):
	t_0 = math.cos((x * -2.0))
	return 1.0 + ((1.0 - t_0) / (-1.0 - t_0))

Julia

function code(x)
	t_0 = cos(Float64(x * -2.0))
	return Float64(1.0 + Float64(Float64(1.0 - t_0) / Float64(-1.0 - t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = cos((x * -2.0));
	tmp = 1.0 + ((1.0 - t_0) / (-1.0 - t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]}, N[(1.0 + N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(-1.0 - t$95$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. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. clear-num N/A

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

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

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

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

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

   7. sqr-cos-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. sqr-sin-a N/A

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

   15. --lowering--.f64 N/A

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

   16. cos-2 N/A

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

   17. cos-sum N/A

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

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

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

   19. cos-lowering-cos.f64 N/A

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

   20. +-lowering-+.f64 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 60.6%

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

8. Taylor expanded in x around inf

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

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

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

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

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

10. Simplified 60.6%

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

11. Final simplification 60.6%

    \[\leadsto 1 + \frac{1 - \cos \left(x \cdot -2\right)}{-1 - \cos \left(x \cdot -2\right)} \]
12. Add Preprocessing

Alternative 6: 59.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. clear-num N/A

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

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

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

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

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

   7. sqr-cos-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. sqr-sin-a N/A

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

   15. --lowering--.f64 N/A

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

   16. cos-2 N/A

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

   17. cos-sum N/A

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

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

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

   19. cos-lowering-cos.f64 N/A

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

   20. +-lowering-+.f64 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 60.6%

   \[\leadsto \frac{1 - \frac{1}{\frac{0.5 + 0.5 \cdot \cos \left(x + x\right)}{0.5 - 0.5 \cdot \cos \left(x + x\right)}}}{\color{blue}{1}} \]
8. Step-by-step derivation

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. count-2 N/A

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

   4. sqr-sin-a N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. frac-times N/A

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

   8. tan-quot N/A

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

   9. tan-quot N/A

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

   10. pow2 N/A

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

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

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

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

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

   13. tan-lowering-tan.f64 60.6

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

9. Applied egg-rr 60.6%

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

Alternative 7: 55.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 + (1.0 / (1.0 / ((0.5 * cos((x + x))) - 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (1.0d0 / (1.0d0 / ((0.5d0 * cos((x + x))) - 0.5d0)))
end function

Java

public static double code(double x) {
	return 1.0 + (1.0 / (1.0 / ((0.5 * Math.cos((x + x))) - 0.5)));
}

Python

def code(x):
	return 1.0 + (1.0 / (1.0 / ((0.5 * math.cos((x + x))) - 0.5)))

Julia

function code(x)
	return Float64(1.0 + Float64(1.0 / Float64(1.0 / Float64(Float64(0.5 * cos(Float64(x + x))) - 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (1.0 / (1.0 / ((0.5 * cos((x + x))) - 0.5)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-times N/A

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

   4. clear-num N/A

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

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

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

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

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

   7. sqr-cos-a N/A

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

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

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

   14. sqr-sin-a N/A

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

   15. --lowering--.f64 N/A

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

   16. cos-2 N/A

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

   17. cos-sum N/A

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

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

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

   19. cos-lowering-cos.f64 N/A

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

   20. +-lowering-+.f64 98.9

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

4. Applied egg-rr 98.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 60.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 56.7%

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

11. Final simplification 56.7%

    \[\leadsto 1 + \frac{1}{\frac{1}{0.5 \cdot \cos \left(x + x\right) - 0.5}} \]
12. 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

4. Applied egg-rr 56.4%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))