Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.1s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (t_0 + 1.0);
}

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) / (t_0 + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (t_0 + 1.0);
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[(t$95$0 + 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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 3: 60.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (tan x) (- (tan x)) 1.0)
  (+
   (/
    1.0
    (/
     (/
      (fma
       (fma
        (fma 0.010582010582010581 (* x x) 0.06666666666666667)
        (* x x)
        -0.6666666666666666)
       (* x x)
       1.0)
      x)
     x))
   1.0)))

C

double code(double x) {
	return fma(tan(x), -tan(x), 1.0) / ((1.0 / ((fma(fma(fma(0.010582010582010581, (x * x), 0.06666666666666667), (x * x), -0.6666666666666666), (x * x), 1.0) / x) / x)) + 1.0);
}

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / Float64(Float64(1.0 / Float64(Float64(fma(fma(fma(0.010582010582010581, Float64(x * x), 0.06666666666666667), Float64(x * x), -0.6666666666666666), Float64(x * x), 1.0) / x) / x)) + 1.0))
end

Wolfram

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $MachinePrecision] / N[(N[(1.0 / N[(N[(N[(N[(N[(0.010582010582010581 * N[(x * x), $MachinePrecision] + 0.06666666666666667), $MachinePrecision] * 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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. unpow-1 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

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

   15. lower-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{{x}^{2}}}}} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{\color{blue}{x \cdot x}}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{x}}{x}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{x}}{x}}}} \]

9. Applied rewrites 59.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}}} \]

10. Final simplification 59.5%

    \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \]
11. Add Preprocessing

Alternative 4: 60.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / 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[(N[Tan[x], $MachinePrecision] * (-N[Tan[x], $MachinePrecision]) + 1.0), $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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. unpow-1 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

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

   15. lower-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 59.4

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

9. Applied rewrites 59.4%

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

10. Final simplification 59.4%

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

Alternative 5: 59.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-pow.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 59.3%

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

Alternative 6: 55.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 55.7%

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

Alternative 7: 55.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (+
   (/
    1.0
    (/
     (/
      (fma
       (fma
        (fma 0.010582010582010581 (* x x) 0.06666666666666667)
        (* x x)
        -0.6666666666666666)
       (* x x)
       1.0)
      x)
     x))
   1.0)))

C

double code(double x) {
	return 1.0 / ((1.0 / ((fma(fma(fma(0.010582010582010581, (x * x), 0.06666666666666667), (x * x), -0.6666666666666666), (x * x), 1.0) / x) / x)) + 1.0);
}

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(1.0 / Float64(Float64(fma(fma(fma(0.010582010582010581, Float64(x * x), 0.06666666666666667), Float64(x * x), -0.6666666666666666), Float64(x * x), 1.0) / x) / x)) + 1.0))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(1.0 / N[(N[(N[(N[(N[(0.010582010582010581 * N[(x * x), $MachinePrecision] + 0.06666666666666667), $MachinePrecision] * 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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.5

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

4. Applied rewrites 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. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. pow-prod-down N/A

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

   9. unpow-1 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

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

   15. lower-pow.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{{x}^{2}}}}} \]
8. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{\color{blue}{x \cdot x}}}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{x}}{x}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{15} + \frac{2}{189} \cdot {x}^{2}\right) - \frac{2}{3}\right)}{x}}{x}}}} \]

9. Applied rewrites 59.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \frac{1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{1}}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{189}, x \cdot x, \frac{1}{15}\right), x \cdot x, \frac{-2}{3}\right), x \cdot x, 1\right)}{x}}{x}}} \]
11. Step-by-step derivation

12. Applied rewrites 55.3%

    \[\leadsto \frac{\color{blue}{1}}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}} \]

13. Final simplification 55.3%

    \[\leadsto \frac{1}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.010582010582010581, x \cdot x, 0.06666666666666667\right), x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \]
14. Add Preprocessing

Alternative 8: 55.6% 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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 55.3%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))