Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.7s

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

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

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 99.7%

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

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.7%

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

   6. pow2 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \frac{1 - \color{blue}{\left(0 + {\tan x}^{2}\right)}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. +-lft-identity 99.7%

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

6. Simplified 99.7%

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

   1. add-sqr-sqrt 99.4%

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

   2. hypot-1-def 99.5%

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

   3. hypot-1-def 99.5%

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

   4. unpow2 99.5%

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

   5. frac-2neg 99.5%

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

   6. div-inv 99.4%

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

   7. unpow2 99.4%

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

   8. hypot-1-def 99.5%

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

   9. hypot-1-def 99.4%

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

   10. add-sqr-sqrt 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. pow2 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. neg-mul-1 99.7%

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

   4. associate-/l* 99.6%

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

   5. metadata-eval 99.6%

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

   6. distribute-neg-in 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. neg-mul-1 99.6%

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

   9. associate-/r* 99.6%

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

   10. metadata-eval 99.6%

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

   11. associate-/l* 99.7%

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

   12. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 55.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}}{1 + {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (/ (tan x) (+ (* x -0.3333333333333333) (/ 1.0 x))))
  (+ 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return (1.0 - (tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + pow(tan(x), 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - (tan(x) / ((x * (-0.3333333333333333d0)) + (1.0d0 / x)))) / (1.0d0 + (tan(x) ** 2.0d0))
end function

Java

public static double code(double x) {
	return (1.0 - (Math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return (1.0 - (math.tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) / Float64(Float64(x * -0.3333333333333333) + Float64(1.0 / x)))) / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - (tan(x) / ((x * -0.3333333333333333) + (1.0 / x)))) / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. tan-quot 99.6%

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

   2. associate-*r/ 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-/l* 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 63.6%

   \[\leadsto \frac{1 - \frac{\tan x}{\color{blue}{-0.3333333333333333 \cdot x + \frac{1}{x}}}}{1 + \tan x \cdot \tan x} \]
8. Step-by-step derivation

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.7%

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

   6. pow2 99.7%

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

9. Applied egg-rr 63.6%

   \[\leadsto \frac{1 - \frac{\tan x}{-0.3333333333333333 \cdot x + \frac{1}{x}}}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
10. Step-by-step derivation

    1. +-lft-identity 99.7%

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

11. Simplified 63.6%

    \[\leadsto \frac{1 - \frac{\tan x}{-0.3333333333333333 \cdot x + \frac{1}{x}}}{1 + \color{blue}{{\tan x}^{2}}} \]

12. Final simplification 63.6%

    \[\leadsto \frac{1 - \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}}{1 + {\tan x}^{2}} \]
13. Add Preprocessing

Alternative 3: 54.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. fma-def 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in x around 0 63.2%

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

6. Final simplification 63.2%

   \[\leadsto \frac{1}{1 + \tan x \cdot \tan x} \]
7. Add Preprocessing

Alternative 4: 54.5% 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 99.7%

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

3. Taylor expanded in x around 0 62.9%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 62.9%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))