Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.4s

Alternatives: 4

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

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

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]]

Derivation

1. Initial program 99.5%

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

Alternative 2: 58.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Simplified 62.9%

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

6. Final simplification 62.9%

   \[\leadsto 1 - \tan x \cdot \tan x \]
7. Add Preprocessing

Alternative 3: 54.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Power[N[Sin[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. Taylor expanded in x around 0

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

5. Simplified 62.9%

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

6. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-cos.f64 62.9

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

8. Simplified 62.9%

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

9. Taylor expanded in x around 0

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

11. Simplified 59.9%

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

12. Taylor expanded in x around inf

    \[\leadsto \color{blue}{1 - {\sin x}^{2}} \]
13. Step-by-step derivation

    1. lower--.f64 N/A

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

    2. lower-pow.f64 N/A

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

    3. lower-sin.f64 59.9

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

14. Simplified 59.9%

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

Alternative 4: 54.1% accurate, 428.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

5. Simplified 59.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))