Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.8s

Alternatives: 6

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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

   2. +-commutative 99.6%

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

   3. add-log-exp 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. log-prod 99.5%

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

   6. metadata-eval 99.5%

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

   7. add-log-exp 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-undefine 99.6%

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

   10. pow2 99.6%

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

   11. fma-undefine 99.6%

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

   12. pow2 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\tan x}^{2} + 1}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

8. Simplified 99.6%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{\mathsf{log1p}\left({\tan x}^{2}\right)}} + -1 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (/ 2.0 (exp (log1p (pow (tan x) 2.0)))) -1.0))

C

double code(double x) {
	return (2.0 / exp(log1p(pow(tan(x), 2.0)))) + -1.0;
}

Java

public static double code(double x) {
	return (2.0 / Math.exp(Math.log1p(Math.pow(Math.tan(x), 2.0)))) + -1.0;
}

Python

def code(x):
	return (2.0 / math.exp(math.log1p(math.pow(math.tan(x), 2.0)))) + -1.0

Julia

function code(x)
	return Float64(Float64(2.0 / exp(log1p((tan(x) ^ 2.0)))) + -1.0)
end

Wolfram

code[x_] := N[(N[(2.0 / N[Exp[N[Log[1 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

   2. +-commutative 99.6%

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

   3. add-log-exp 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. log-prod 99.5%

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

   6. metadata-eval 99.5%

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

   7. add-log-exp 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-undefine 99.6%

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

   10. pow2 99.6%

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

   11. fma-undefine 99.6%

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

   12. pow2 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\tan x}^{2} + 1}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

8. Simplified 99.6%

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

   1. div-sub 99.4%

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

   2. pow2 99.4%

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

   3. sub-neg 99.4%

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

   4. pow2 99.4%

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

10. Applied egg-rr 99.4%

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

    1. sub-neg 99.4%

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

    2. div-sub 99.6%

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

    3. +-lft-identity 99.6%

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

    4. metadata-eval 99.6%

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

    5. associate-+r+ 99.5%

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

    6. associate--r+ 99.5%

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

    7. metadata-eval 99.5%

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

    8. div-sub 99.4%

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

12. Simplified 99.2%

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

    1. hypot-undefine 99.1%

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

    2. metadata-eval 99.1%

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

    3. unpow2 99.1%

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

    4. sqrt-pow2 99.4%

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

    5. metadata-eval 99.4%

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

    6. pow1 99.4%

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

    7. add-exp-log 99.4%

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

    8. log1p-define 99.4%

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

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

    \[\leadsto \frac{2}{e^{\mathsf{log1p}\left({\tan x}^{2}\right)}} + -1 \]
16. Add Preprocessing

Alternative 3: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

   2. +-commutative 99.6%

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

   3. add-log-exp 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. log-prod 99.5%

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

   6. metadata-eval 99.5%

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

   7. add-log-exp 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-undefine 99.6%

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

   10. pow2 99.6%

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

   11. fma-undefine 99.6%

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

   12. pow2 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\tan x}^{2} + 1}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

8. Simplified 99.6%

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

   1. div-sub 99.4%

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

   2. pow2 99.4%

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

   3. sub-neg 99.4%

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

   4. pow2 99.4%

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

10. Applied egg-rr 99.4%

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

    1. sub-neg 99.4%

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

    2. div-sub 99.6%

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

    3. +-lft-identity 99.6%

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

    4. metadata-eval 99.6%

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

    5. associate-+r+ 99.5%

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

    6. associate--r+ 99.5%

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

    7. metadata-eval 99.5%

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

    8. div-sub 99.4%

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

12. Simplified 99.2%

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

    1. hypot-undefine 99.1%

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

    2. metadata-eval 99.1%

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

    3. unpow2 99.1%

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

    4. sqrt-pow2 99.4%

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

    5. metadata-eval 99.4%

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

    6. pow1 99.4%

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

    7. +-commutative 99.4%

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

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

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

Alternative 4: 59.8% 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 99.6%

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

3. Taylor expanded in x around 0 63.1%

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

4. Final simplification 63.1%

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

Alternative 5: 55.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

   2. +-commutative 99.6%

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

   3. add-log-exp 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. log-prod 99.5%

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

   6. metadata-eval 99.5%

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

   7. add-log-exp 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-undefine 99.6%

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

   10. pow2 99.6%

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

   11. fma-undefine 99.6%

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

   12. pow2 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\tan x}^{2} + 1}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

8. Simplified 99.6%

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

   1. div-sub 99.4%

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

   2. pow2 99.4%

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

   3. sub-neg 99.4%

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

   4. pow2 99.4%

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

10. Applied egg-rr 99.4%

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

    1. sub-neg 99.4%

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

    2. div-sub 99.6%

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

    3. +-lft-identity 99.6%

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

    4. metadata-eval 99.6%

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

    5. associate-+r+ 99.5%

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

    6. associate--r+ 99.5%

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

    7. metadata-eval 99.5%

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

    8. div-sub 99.4%

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

12. Simplified 99.2%

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

13. Taylor expanded in x around 0 59.9%

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

    1. +-commutative 59.9%

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

    2. unpow2 59.9%

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

    3. fma-define 59.9%

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

15. Simplified 59.9%

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

16. Final simplification 59.9%

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

Alternative 6: 55.6% 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 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 99.6%

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

   2. +-commutative 99.6%

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

   3. add-log-exp 99.5%

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

   4. *-un-lft-identity 99.5%

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

   5. log-prod 99.5%

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

   6. metadata-eval 99.5%

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

   7. add-log-exp 99.6%

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

   8. +-commutative 99.6%

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

   9. fma-undefine 99.6%

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

   10. pow2 99.6%

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

   11. fma-undefine 99.6%

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

   12. pow2 99.6%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1 - {\tan x}^{2}}{{\tan x}^{2} + 1}} \]
7. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in x around 0 58.9%

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

Reproduce

herbie shell --seed 2024147 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))