Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

Alternatives: 4

Speedup: N/A×

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

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

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

4. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 59.1% 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 99.5%

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

4. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. frac-2neg 99.5%

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

   2. pow2 99.5%

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

   3. div-inv 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0 58.4%

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

10. Final simplification 58.4%

    \[\leadsto 1 - {\tan x}^{2} \]
11. Add Preprocessing

Alternative 3: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - {\tan x}^{4} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (pow (tan x) 4.0)))

C

double code(double x) {
	return 1.0 - pow(tan(x), 4.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (tan(x) ** 4.0d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.pow(Math.tan(x), 4.0);
}

Python

def code(x):
	return 1.0 - math.pow(math.tan(x), 4.0)

Julia

function code(x)
	return Float64(1.0 - (tan(x) ^ 4.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (tan(x) ^ 4.0);
end

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 4.0], $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

4. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. div-sub 99.3%

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

   2. pow2 99.3%

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

   3. add-exp-log 99.3%

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

   4. log1p-undefine 99.3%

      \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left({\tan x}^{2}\right)}}} - \frac{{\tan x}^{2}}{1 + \tan x \cdot \tan x} \]

   5. exp-neg 99.3%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left({\tan x}^{2}\right)}} - \frac{{\tan x}^{2}}{1 + \tan x \cdot \tan x} \]

   6. sub-neg 99.3%

      \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left({\tan x}^{2}\right)} + \left(-\frac{{\tan x}^{2}}{1 + \tan x \cdot \tan x}\right)} \]

8. Applied egg-rr 57.6%

   \[\leadsto \color{blue}{\left({\tan x}^{2} + 1\right) + \left(-\left({\tan x}^{2} + {\tan x}^{4}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 57.6%

      \[\leadsto \color{blue}{\left(-\left({\tan x}^{2} + {\tan x}^{4}\right)\right) + \left({\tan x}^{2} + 1\right)} \]

   2. associate-+r+ 57.6%

      \[\leadsto \color{blue}{\left(\left(-\left({\tan x}^{2} + {\tan x}^{4}\right)\right) + {\tan x}^{2}\right) + 1} \]

   3. metadata-eval 57.6%

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

   4. pow-sqr 57.6%

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

   5. distribute-rgt1-in 57.6%

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

   6. distribute-rgt-neg-out 57.6%

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

   7. remove-double-neg 57.6%

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

   8. neg-mul-1 57.6%

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

   9. distribute-rgt-in 57.6%

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

   10. associate-+l+ 57.6%

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

   11. metadata-eval 57.6%

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

   12. +-rgt-identity 57.6%

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

   13. distribute-lft-neg-in 57.6%

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

   14. pow-sqr 57.6%

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

   15. metadata-eval 57.6%

       \[\leadsto \left(-{\tan x}^{\color{blue}{4}}\right) + 1 \]

10. Simplified 57.6%

    \[\leadsto \color{blue}{1 - {\tan x}^{4}} \]
11. Add Preprocessing

Alternative 4: 54.8% 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 99.5%

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

4. Applied egg-rr 99.5%

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

   1. unpow2 99.5%

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

6. Simplified 99.5%

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

   1. pow2 99.5%

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

   2. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in x around 0 53.5%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))