Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.3s

Alternatives: 3

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 99.4%

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

   1. add-log-exp 99.2%

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

   2. *-un-lft-identity 99.2%

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

   3. log-prod 99.2%

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

   4. metadata-eval 99.2%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

   7. add-sqr-sqrt 99.2%

      \[\leadsto 0 + \frac{1 - {\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   8. pow2 99.2%

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

   9. hypot-1-def 99.2%

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

4. Applied egg-rr 99.2%

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

   1. +-lft-identity 99.2%

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

   2. unpow2 99.2%

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

   3. hypot-undefine 99.2%

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

   4. metadata-eval 99.2%

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

   5. unpow2 99.2%

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

   6. rem-exp-log 99.2%

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

   7. log1p-undefine 99.2%

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

   8. hypot-undefine 99.2%

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

   9. metadata-eval 99.2%

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

   10. unpow2 99.2%

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

   11. rem-exp-log 99.1%

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

   12. log1p-undefine 99.1%

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

   13. rem-square-sqrt 99.3%

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

   14. log1p-undefine 99.3%

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

   15. rem-exp-log 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 59.2% 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 99.4%

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

   1. +-commutative 99.4%

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

   2. fma-define 99.4%

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

3. Simplified 99.4%

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

   1. add-log-exp 98.6%

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

   2. *-un-lft-identity 98.6%

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

   3. log-prod 98.6%

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

   4. metadata-eval 98.6%

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

   5. add-log-exp 99.4%

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

   6. pow2 99.4%

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

6. Applied egg-rr 99.4%

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

   1. +-lft-identity 99.4%

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

8. Simplified 99.4%

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

   1. flip-- 99.2%

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

   2. metadata-eval 99.2%

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

   3. pow-sqr 99.2%

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

   4. metadata-eval 99.2%

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

   5. fma-undefine 99.1%

      \[\leadsto \frac{\frac{1 - {\tan x}^{4}}{1 + {\tan x}^{2}}}{\color{blue}{\tan x \cdot \tan x + 1}} \]

   6. unpow2 99.1%

      \[\leadsto \frac{\frac{1 - {\tan x}^{4}}{1 + {\tan x}^{2}}}{\color{blue}{{\tan x}^{2}} + 1} \]

   7. +-commutative 99.1%

      \[\leadsto \frac{\frac{1 - {\tan x}^{4}}{1 + {\tan x}^{2}}}{\color{blue}{1 + {\tan x}^{2}}} \]

   8. associate-/r* 99.1%

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

   9. pow2 99.1%

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

   10. metadata-eval 99.1%

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

   11. sqrt-pow2 98.9%

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

   12. metadata-eval 98.9%

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

   13. unpow2 98.9%

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

   14. hypot-undefine 98.9%

       \[\leadsto \frac{1 - {\tan x}^{4}}{{\color{blue}{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}}^{4}} \]

   15. frac-2neg 98.9%

       \[\leadsto \color{blue}{\frac{-\left(1 - {\tan x}^{4}\right)}{-{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{4}}} \]

10. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{\left(-\left(1 - {\tan x}^{4}\right)\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{4}}} \]

11. Taylor expanded in x around 0 54.9%

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

12. Final simplification 54.9%

    \[\leadsto 1 - {\tan x}^{4} \]
13. Add Preprocessing

Alternative 3: 55.8% 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 99.4%

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

3. Taylor expanded in x around 0 51.4%

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

4. Final simplification 51.4%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024042 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))