Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 8.3s

Alternatives: 9

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 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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 99.5

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
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 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 99.3

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

   10. lift-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.3

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

   14. lift-/.f64 N/A

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

   15. frac-2neg N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}}} \]
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 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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 99.4

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

6. Applied rewrites 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

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 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-cos.f64 99.3

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

4. Applied rewrites 99.3%

   \[\leadsto \frac{1 - \color{blue}{\sin x \cdot \frac{\tan x}{\cos x}}}{1 + \tan x \cdot \tan x} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.3

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 99.3

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

6. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. tan-quot N/A

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

   8. lift-tan.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 99.4

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

8. Applied rewrites 99.4%

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

Alternative 5: 59.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (pow (tan x) 2.0))
  (+
   (pow
    (/
     (fma
      (fma
       (fma -0.0021164021164021165 (* x x) -0.022222222222222223)
       (* x x)
       -0.3333333333333333)
      (* x x)
      1.0)
     x)
    -2.0)
   1.0)))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / (pow((fma(fma(fma(-0.0021164021164021165, (x * x), -0.022222222222222223), (x * x), -0.3333333333333333), (x * x), 1.0) / x), -2.0) + 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / Float64((Float64(fma(fma(fma(-0.0021164021164021165, Float64(x * x), -0.022222222222222223), Float64(x * x), -0.3333333333333333), Float64(x * x), 1.0) / x) ^ -2.0) + 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(N[(N[(-0.0021164021164021165 * N[(x * x), $MachinePrecision] + -0.022222222222222223), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\color{blue}{\left(\frac{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{x}\right)}}^{-2}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{x}\right)}^{-2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) \cdot {x}^{2}} + 1}{x}\right)}^{-2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) - \frac{1}{3}, {x}^{2}, 1\right)}}{x}\right)}^{-2}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945} \cdot {x}^{2} - \frac{1}{45}, {x}^{2}, \frac{-1}{3}\right)}, {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   9. sub-neg N/A

      \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945} \cdot {x}^{2} + \color{blue}{\frac{-1}{45}}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945}, {x}^{2}, \frac{-1}{45}\right)}, {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{x \cdot x}, \frac{-1}{45}\right), {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{x \cdot x}, \frac{-1}{45}\right), {x}^{2}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   14. unpow2 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), \color{blue}{x \cdot x}, \frac{-1}{3}\right), {x}^{2}, 1\right)}{x}\right)}^{-2}} \]

   16. unpow2 N/A

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, x \cdot x, \frac{-1}{45}\right), x \cdot x, \frac{-1}{3}\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

   17. lower-*.f64 63.2

       \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right)}{x}\right)}^{-2}} \]

9. Applied rewrites 63.2%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\color{blue}{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}}^{-2}} \]

10. Final simplification 63.2%

    \[\leadsto \frac{1 - {\tan x}^{2}}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, x \cdot x, -0.022222222222222223\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right)}{x}\right)}^{-2} + 1} \]
11. Add Preprocessing

Alternative 6: 59.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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. pow2 N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. clear-num N/A

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

   12. tan-quot N/A

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

   13. lift-tan.f64 N/A

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

   14. inv-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 63.2

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

9. Applied rewrites 63.2%

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

10. Final simplification 63.2%

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

Alternative 7: 58.6% 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 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 63.1%

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

Alternative 8: 54.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
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 99.4

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. associate-*l/ N/A

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

   7. associate-*r/ N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 99.3

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

   10. lift-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-+.f64 99.3

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

   14. lift-/.f64 N/A

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

   15. frac-2neg N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 59.7%

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

Alternative 9: 54.3% 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 99.4%

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

4. Applied rewrites 59.4%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))