Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.8s

Alternatives: 6

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-define 99.5%

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

4. Applied egg-rr 99.5%

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

   1. add-log-exp 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. log-prod 98.4%

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

   4. metadata-eval 98.4%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. add-log-exp 98.4%

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

   2. *-un-lft-identity 98.4%

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

   3. log-prod 98.4%

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

   4. metadata-eval 98.4%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 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

   1. add-log-exp 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. log-prod 99.3%

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

   4. metadata-eval 99.3%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

   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 99.3%

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

4. Applied egg-rr 99.3%

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

   1. +-lft-identity 99.3%

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

   2. unpow2 99.3%

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

   3. hypot-undefine 99.4%

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

   4. metadata-eval 99.4%

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

   5. unpow2 99.4%

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

   6. rem-exp-log 99.3%

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

   7. log1p-undefine 99.3%

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

   8. hypot-undefine 99.3%

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

   9. metadata-eval 99.3%

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

   10. unpow2 99.3%

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

   11. rem-exp-log 99.2%

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

   12. log1p-undefine 99.2%

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

   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 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-define 99.5%

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.5%

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

   1. distribute-neg-frac 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-neg-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. neg-sub0 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate--r+ 99.4%

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

   8. associate-+r- 99.4%

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

   9. metadata-eval 99.4%

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

   10. div-sub 99.3%

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

   11. rem-exp-log 99.0%

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

   12. log1p-undefine 99.0%

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

   13. rem-exp-log 99.2%

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

   14. log1p-undefine 99.3%

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

   15. exp-diff 99.3%

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

8. Simplified 99.3%

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

   1. pow2 99.3%

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

   2. +-commutative 99.3%

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

   3. fma-define 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 5: 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 99.5%

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-define 99.5%

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.5%

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

   1. distribute-neg-frac 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-neg-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. neg-sub0 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate--r+ 99.4%

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

   8. associate-+r- 99.4%

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

   9. metadata-eval 99.4%

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

   10. div-sub 99.3%

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

   11. rem-exp-log 99.0%

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

   12. log1p-undefine 99.0%

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

   13. rem-exp-log 99.2%

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

   14. log1p-undefine 99.3%

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

   15. exp-diff 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 6: 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. Taylor expanded in x around 0 54.3%

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

4. Final simplification 54.3%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))