Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 5.4s

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-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

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: 57.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.17:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.17)
   (/ -1.0 (- -1.0 (pow (tan x) 2.0)))
   (/ (- 1.0 (* x x)) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.17) {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	} else {
		tmp = (1.0 - (x * x)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.17)
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * x)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.17], N[(-1.0 / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1.1699999999999999

   1. Initial program 99.6%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. distribute-frac-neg2 N/A

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

      15. lower-/.f64 N/A

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

   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 74.1%

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

   if 1.1699999999999999 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 9.7

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

   8. Applied rewrites 9.7%

      \[\leadsto \frac{1 - x \cdot x}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 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

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

   2. lift--.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. div-sub N/A

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

   8. frac-sub N/A

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

6. Applied rewrites 99.5%

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

Alternative 4: 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-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-frac-neg N/A

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

   14. distribute-frac-neg2 N/A

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

   15. lower-/.f64 N/A

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

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. pow2 N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. lower--.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 5: 56.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1.17:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (tan x) (tan x)) 1.17) 1.0 (/ (- 1.0 (* x x)) (fma x x 1.0))))

C

double code(double x) {
	double tmp;
	if ((tan(x) * tan(x)) <= 1.17) {
		tmp = 1.0;
	} else {
		tmp = (1.0 - (x * x)) / fma(x, x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 1.17)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 - Float64(x * x)) / fma(x, x, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 1.17], 1.0, N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 1.1699999999999999

   1. Initial program 99.6%

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

   4. Applied rewrites 73.7%

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

   if 1.1699999999999999 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 9.7

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

   8. Applied rewrites 9.7%

      \[\leadsto \frac{1 - x \cdot x}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 58.6%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{1}} \]
6. Add Preprocessing

Alternative 7: 54.7% 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 54.0%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))