Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 6.1s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 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 99.6

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

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. remove-double-neg N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. frac-2neg N/A

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

   8. remove-double-neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-in N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 N/A

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

   16. remove-double-neg N/A

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

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 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 99.6

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

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

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

Alternative 4: 60.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (/
   (-
    -1.0
    (pow
     (/
      1.0
      (/
       (fma
        (fma -0.022222222222222223 (* x x) -0.3333333333333333)
        (* x x)
        1.0)
       x))
     2.0))
   (fma (tan x) (tan x) -1.0))))

C

double code(double x) {
	return 1.0 / ((-1.0 - pow((1.0 / (fma(fma(-0.022222222222222223, (x * x), -0.3333333333333333), (x * x), 1.0) / x)), 2.0)) / fma(tan(x), tan(x), -1.0));
}

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(-1.0 - (Float64(1.0 / Float64(fma(fma(-0.022222222222222223, Float64(x * x), -0.3333333333333333), Float64(x * x), 1.0) / x)) ^ 2.0)) / fma(tan(x), tan(x), -1.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(-1.0 - N[Power[N[(1.0 / N[(N[(N[(-0.022222222222222223 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $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 99.6

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

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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fma.f64 N/A

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

6. Applied rewrites 99.5%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right)}{x}}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right)}{x}}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right) + 1}}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    3. *-commutative N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\color{blue}{\left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}\right) \cdot {x}^{2}} + 1}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    4. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {x}^{2} - \frac{1}{3}, {x}^{2}, 1\right)}}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    5. sub-neg N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {x}^{2}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, {x}^{2}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    7. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {x}^{2}, \frac{-1}{3}\right)}, {x}^{2}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    8. unpow2 N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    10. unpow2 N/A

        \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, x \cdot x, \frac{-1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

    11. lower-*.f64 61.9

        \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]

11. Applied rewrites 61.9%

    \[\leadsto \frac{1}{\frac{-1 - {\left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}}}\right)}^{2}}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}} \]
12. Add Preprocessing

Alternative 5: 60.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)}{x}}, \tan x, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (tan x) (- (tan x)) 1.0)
  (fma (/ 1.0 (/ (fma -0.3333333333333333 (* x x) 1.0) x)) (tan x) 1.0)))

C

double code(double x) {
	return fma(tan(x), -tan(x), 1.0) / fma((1.0 / (fma(-0.3333333333333333, (x * x), 1.0) / x)), tan(x), 1.0);
}

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / fma(Float64(1.0 / Float64(fma(-0.3333333333333333, Float64(x * x), 1.0) / x)), tan(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[(-0.3333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $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 99.6

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 61.8

       \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{x \cdot x}, 1\right)}{x}}, \tan x, 1\right)} \]

11. Applied rewrites 61.8%

    \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{\mathsf{fma}\left(\frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)}{x}}}, \tan x, 1\right)} \]
12. Add Preprocessing

Alternative 6: 60.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (pow (tan x) 2.0))
  (+
   (/
    1.0
    (/
     (/
      (fma (fma 0.06666666666666667 (* x x) -0.6666666666666666) (* x x) 1.0)
      x)
     x))
   1.0)))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / ((1.0 / ((fma(fma(0.06666666666666667, (x * x), -0.6666666666666666), (x * x), 1.0) / x) / x)) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / 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[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. associate-*l/ N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 99.4

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

4. Applied rewrites 99.4%

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

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 61.8

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

7. Applied rewrites 61.8%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-pow.f64 61.8

      \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}} \]

9. Applied rewrites 61.8%

   \[\leadsto \frac{1 - \color{blue}{{\tan x}^{2}}}{1 + \frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}}} \]

10. Final simplification 61.8%

    \[\leadsto \frac{1 - {\tan x}^{2}}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.06666666666666667, x \cdot x, -0.6666666666666666\right), x \cdot x, 1\right)}{x}}{x}} + 1} \]
11. Add Preprocessing

Alternative 7: 59.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $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 99.6

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

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. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. frac-2neg N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. remove-double-neg N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lift-fma.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 61.6%

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

Alternative 8: 59.9% 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 61.6%

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

Alternative 9: 56.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / 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 99.6

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

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

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 58.1%

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

Alternative 10: 55.7% 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 57.7%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))