Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 7.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 99.5

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift-+.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{t\_0 - 1}{-1 - t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- t_0 1.0) (- -1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (t_0 - 1.0) / (-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 = (t_0 - 1.0d0) / ((-1.0d0) - t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (t_0 - 1.0) / (-1.0 - t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (t_0 - 1.0) / (-1.0 - t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(t_0 - 1.0) / Float64(-1.0 - t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (t_0 - 1.0) / (-1.0 - t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(t$95$0 - 1.0), $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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 99.5

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift-+.f64 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 59.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \tan x \cdot \tan x}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (* (tan x) (tan x)))
  (+
   (/
    1.0
    (/
     (/
      (fma (fma 0.06666666666666667 (* x x) -0.6666666666666666) (* x x) 1.0)
      x)
     x))
   1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / ((1.0 / ((fma(fma(0.06666666666666667, (x * x), -0.6666666666666666), (x * x), 1.0) / x) / x)) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(Float64(1.0 / Float64(Float64(fma(fma(0.06666666666666667, Float64(x * x), -0.6666666666666666), Float64(x * x), 1.0) / x) / x)) + 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(N[(N[(N[(0.06666666666666667 * N[(x * x), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-times N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. clear-num N/A

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

   14. tan-quot N/A

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

   15. lift-tan.f64 N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{{x}^{2}}}}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{\color{blue}{x \cdot x}}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{x}}{x}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{x}}{x}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right)}{x}}}{x}}} \]

   5. +-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right) + 1}}{x}}{x}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\color{blue}{\left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}\right) \cdot {x}^{2}} + 1}{x}}{x}}} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{15} \cdot {x}^{2} - \frac{2}{3}, {x}^{2}, 1\right)}}{x}}{x}}} \]

   8. sub-neg N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, {x}^{2}, 1\right)}{x}}{x}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\frac{1}{15} \cdot {x}^{2} + \color{blue}{\frac{-2}{3}}, {x}^{2}, 1\right)}{x}}{x}}} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{15}, {x}^{2}, \frac{-2}{3}\right)}, {x}^{2}, 1\right)}{x}}{x}}} \]

   11. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{15}, \color{blue}{x \cdot x}, \frac{-2}{3}\right), {x}^{2}, 1\right)}{x}}{x}}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{15}, \color{blue}{x \cdot x}, \frac{-2}{3}\right), {x}^{2}, 1\right)}{x}}{x}}} \]

   13. unpow2 N/A

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{15}, x \cdot x, \frac{-2}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}}{x}}} \]

   14. lower-*.f64 61.5

       \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), \color{blue}{x \cdot x}, 1\right)}{x}}{x}}} \]

7. Applied rewrites 61.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \frac{1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}}} \]

8. Final simplification 61.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \]
9. Add Preprocessing

Alternative 5: 59.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 99.5

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift-+.f64 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 99.5

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 61.1%

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

Alternative 6: 55.2% 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 99.5

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-in N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-fma.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

   16. neg-sub0 N/A

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

   17. lift-+.f64 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 57.4%

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

Alternative 7: 54.8% accurate, 428.0× 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

4. Applied rewrites 57.0%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))