Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 9.9s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 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), 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. Step-by-step derivation

   1. frac-2neg 99.5%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. fma-udef 99.5%

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

   9. fma-udef 99.5%

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

   10. neg-mul-1 99.5%

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

   11. +-commutative 99.5%

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

   12. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

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. Step-by-step derivation

   1. pow2 99.5%

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

   2. metadata-eval 99.5%

      \[\leadsto \frac{1 - {\tan x}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}}}{1 + \tan x \cdot \tan x} \]

   3. metadata-eval 99.5%

      \[\leadsto \frac{1 - {\tan x}^{\left(\color{blue}{\left(2 \cdot 3\right)} \cdot 0.3333333333333333\right)}}{1 + \tan x \cdot \tan x} \]

   4. pow-pow 99.2%

      \[\leadsto \frac{1 - \color{blue}{{\left({\tan x}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}}}{1 + \tan x \cdot \tan x} \]

   5. metadata-eval 99.2%

      \[\leadsto \frac{1 - {\left({\tan x}^{\color{blue}{6}}\right)}^{0.3333333333333333}}{1 + \tan x \cdot \tan x} \]

3. Applied egg-rr 99.2%

   \[\leadsto \frac{1 - \color{blue}{{\left({\tan x}^{6}\right)}^{0.3333333333333333}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. pow-pow 99.5%

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{\left(6 \cdot 0.3333333333333333\right)}}}{1 + \tan x \cdot \tan x} \]

   2. metadata-eval 99.5%

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

   3. pow2 99.5%

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

   4. tan-quot 99.4%

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

   5. associate-*r/ 99.4%

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

   6. expm1-log1p-u 99.2%

      \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan x \cdot \sin x}{\cos x}\right)\right)}}{1 + \tan x \cdot \tan x} \]

   7. expm1-udef 99.0%

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

   8. log1p-udef 99.1%

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

   9. add-exp-log 99.3%

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

   10. associate-*r/ 99.3%

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

   11. tan-quot 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

   2. fma-udef 99.4%

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

   3. unpow2 99.4%

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

   4. +-commutative 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. div-sub 99.2%

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

   2. pow2 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. +-commutative 99.2%

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

   6. pow2 99.2%

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

   7. +-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. div-sub 99.4%

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

    2. associate--r+ 99.4%

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

    3. metadata-eval 99.4%

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

    4. +-commutative 99.4%

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

    5. associate--r+ 99.5%

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

    6. metadata-eval 99.5%

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

    7. +-commutative 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.0 + N[(2.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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. pow2 99.5%

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

   2. metadata-eval 99.5%

      \[\leadsto \frac{1 - {\tan x}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}}}{1 + \tan x \cdot \tan x} \]

   3. metadata-eval 99.5%

      \[\leadsto \frac{1 - {\tan x}^{\left(\color{blue}{\left(2 \cdot 3\right)} \cdot 0.3333333333333333\right)}}{1 + \tan x \cdot \tan x} \]

   4. pow-pow 99.2%

      \[\leadsto \frac{1 - \color{blue}{{\left({\tan x}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333}}}{1 + \tan x \cdot \tan x} \]

   5. metadata-eval 99.2%

      \[\leadsto \frac{1 - {\left({\tan x}^{\color{blue}{6}}\right)}^{0.3333333333333333}}{1 + \tan x \cdot \tan x} \]

3. Applied egg-rr 99.2%

   \[\leadsto \frac{1 - \color{blue}{{\left({\tan x}^{6}\right)}^{0.3333333333333333}}}{1 + \tan x \cdot \tan x} \]
4. Step-by-step derivation

   1. pow-pow 99.5%

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{\left(6 \cdot 0.3333333333333333\right)}}}{1 + \tan x \cdot \tan x} \]

   2. metadata-eval 99.5%

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

   3. pow2 99.5%

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

   4. tan-quot 99.4%

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

   5. associate-*r/ 99.4%

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

   6. expm1-log1p-u 99.2%

      \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan x \cdot \sin x}{\cos x}\right)\right)}}{1 + \tan x \cdot \tan x} \]

   7. expm1-udef 99.0%

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

   8. log1p-udef 99.1%

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

   9. add-exp-log 99.3%

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

   10. associate-*r/ 99.3%

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

   11. tan-quot 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sub-neg 99.4%

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

   2. fma-udef 99.4%

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

   3. unpow2 99.4%

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

   4. +-commutative 99.4%

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

   5. metadata-eval 99.4%

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

7. Simplified 99.4%

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

   1. div-sub 99.2%

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

   2. pow2 99.2%

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

   3. +-commutative 99.2%

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

   4. +-commutative 99.2%

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

   5. +-commutative 99.2%

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

   6. pow2 99.2%

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

   7. +-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

    1. div-sub 99.4%

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

    2. associate--r+ 99.4%

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

    3. metadata-eval 99.4%

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

    4. div-sub 99.3%

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

    5. sub-neg 99.3%

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

    6. +-commutative 99.3%

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

    7. *-inverses 99.3%

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

    8. metadata-eval 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 5: 55.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.0 / 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. Step-by-step derivation

   1. frac-2neg 99.5%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. fma-udef 99.5%

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

   9. fma-udef 99.5%

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

   10. neg-mul-1 99.5%

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

   11. +-commutative 99.5%

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

   12. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 54.6%

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

7. Final simplification 54.6%

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

Alternative 6: 54.9% 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. Taylor expanded in x around 0 54.2%

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

3. Final simplification 54.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023181 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))