Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 5.1s

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, 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(Float64(-fma(tan(x), 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

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $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 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. pow2 N/A

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

   3. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (t_0 + 1.0d0)
end function

Java

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

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (t_0 + 1.0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (t_0 + 1.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 59.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $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. Taylor expanded in x around 0

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

5. Applied rewrites 57.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{1}} \]
6. Add Preprocessing

Alternative 5: 55.6% 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(1.0 / Float64((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[(1.0 / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 53.3%

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

12. Final simplification 53.3%

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

Alternative 6: 55.2% 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 rewrites 52.9%

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

Reproduce

herbie shell --seed 2024327 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))