2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.8%

Time: 14.7s

Alternatives: 15

Speedup: 17.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan x, \sin x, \cos x\right)\\ \frac{\mathsf{fma}\left(t\_0, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right), t\_0\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (tan x) (sin x) (cos x))))
   (/
    (* (fma t_0 (* 0.3333333333333333 (* eps eps)) t_0) eps)
    (* (fma (- (tan x)) (tan eps) 1.0) (cos x)))))

C

double code(double x, double eps) {
	double t_0 = fma(tan(x), sin(x), cos(x));
	return (fma(t_0, (0.3333333333333333 * (eps * eps)), t_0) * eps) / (fma(-tan(x), tan(eps), 1.0) * cos(x));
}

Julia

function code(x, eps)
	t_0 = fma(tan(x), sin(x), cos(x))
	return Float64(Float64(fma(t_0, Float64(0.3333333333333333 * Float64(eps * eps)), t_0) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[((-N[Tan[x], $MachinePrecision]) * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333 \cdot \mathsf{fma}\left(\sin x, \frac{\sin x}{\cos x}, \cos x\right), \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{\cos x}, \cos x\right)\right) \cdot \varepsilon}}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]
8. Step-by-step derivation

9. Applied rewrites 99.9%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \sin x, \cos x\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(\tan x, \sin x, \cos x\right)\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(Float64(fma(0.3333333333333333, Float64(eps * eps), fma(sin(x), Float64(sin(x) / cos(x)), cos(x))) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * N[(eps * eps), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] / N[(N[((-N[Tan[x], $MachinePrecision]) * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \varepsilon \cdot \varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{\cos x}, \cos x\right)\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]
9. Step-by-step derivation

10. Applied rewrites 99.9%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-cos.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\\ t_1 := t\_0 + \frac{\mathsf{PI}\left(\right)}{2}\\ \mathsf{fma}\left(\frac{{\sin x}^{2}}{\frac{\sin \left(t\_1 - t\_0\right) + \sin \left(t\_1 + t\_0\right)}{2}}, \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (+ (PI) x) (PI))) (t_1 (+ t_0 (/ (PI) 2.0))))
   (fma
    (/ (pow (sin x) 2.0) (/ (+ (sin (- t_1 t_0)) (sin (+ t_1 t_0))) 2.0))
    eps
    eps)))

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{\frac{\sin \left(\left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) - \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right) + \sin \left(\left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) + \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}{2}}, \varepsilon, \varepsilon\right) \]
8. Add Preprocessing

Alternative 5: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/
   (pow (sin x) 2.0)
   (- 0.5 (* 0.5 (cos (* 2.0 (+ (+ (+ (PI) x) (PI)) (/ (PI) 2.0)))))))
  eps
  eps))

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}, \varepsilon, \varepsilon\right) \]
8. Add Preprocessing

Alternative 6: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/ (pow (sin x) 2.0) (+ 0.5 (* 0.5 (cos (* 2.0 (+ (+ (PI) x) (PI)))))))
  eps
  eps))

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
8. Add Preprocessing

Alternative 7: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\tan x \cdot \sin x}{\cos x}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (/ (* (tan x) (sin x)) (cos x)) eps eps))

C

double code(double x, double eps) {
	return fma(((tan(x) * sin(x)) / cos(x)), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(tan(x) * sin(x)) / cos(x)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{\tan x \cdot \sin x}{\cos x}, \varepsilon, \varepsilon\right) \]
8. Add Preprocessing

Alternative 8: 99.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (fma (pow (tan x) 2.0) eps eps))

C

double code(double x, double eps) {
	return fma(pow(tan(x), 2.0), eps, eps);
}

Julia

function code(x, eps)
	return fma((tan(x) ^ 2.0), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)} \]
8. Add Preprocessing

Alternative 9: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0031746031746031746, x \cdot x, 0.044444444444444446\right) \cdot x\right) \cdot x - 0.3333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/
   (*
    (fma
     (-
      (* (* (fma -0.0031746031746031746 (* x x) 0.044444444444444446) x) x)
      0.3333333333333333)
     (* x x)
     1.0)
    (* x x))
   (+ 0.5 (* 0.5 (cos (* 2.0 (+ (+ (PI) x) (PI)))))))
  eps
  eps))

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{45} + \frac{-1}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.0031746031746031746, x \cdot x, 0.044444444444444446\right) \cdot x\right) \cdot x - 0.3333333333333333, x \cdot x, 1\right) \cdot \left(x \cdot x\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 10: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/
   (*
    (*
     (fma (- (* 0.044444444444444446 (* x x)) 0.3333333333333333) (* x x) 1.0)
     x)
    x)
   (+ 0.5 (* 0.5 (cos (* 2.0 (+ (+ (PI) x) (PI)))))))
  eps
  eps))

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 11: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/
   (*
    (*
     (fma (- (* 0.044444444444444446 (* x x)) 0.3333333333333333) (* x x) 1.0)
     x)
    x)
   (+ 0.5 (* 0.5 (cos (+ x x)))))
  eps
  eps))

C

double code(double x, double eps) {
	return fma((((fma(((0.044444444444444446 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x) * x) / (0.5 + (0.5 * cos((x + x))))), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(Float64(fma(Float64(Float64(0.044444444444444446 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x) * x) / Float64(0.5 + Float64(0.5 * cos(Float64(x + x))))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{{\sin x}^{2}}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{45} \cdot {x}^{2} - \frac{1}{3}\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\mathsf{PI}\left(\right) + x\right) + \mathsf{PI}\left(\right)\right)\right)}, \varepsilon, \varepsilon\right) \]

12. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 12: 98.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (*
    (fma
     (fma
      (fma 0.19682539682539682 (* x x) 0.37777777777777777)
      (* x x)
      0.6666666666666666)
     (* x x)
     1.0)
    x)
   x)
  eps
  eps))

C

double code(double x, double eps) {
	return fma(((fma(fma(fma(0.19682539682539682, (x * x), 0.37777777777777777), (x * x), 0.6666666666666666), (x * x), 1.0) * x) * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(fma(fma(fma(0.19682539682539682, Float64(x * x), 0.37777777777777777), Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * x) * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(0.19682539682539682 * N[(x * x), $MachinePrecision] + 0.37777777777777777), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 13: 98.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (* (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0) x)
   x)
  eps
  eps))

C

double code(double x, double eps) {
	return fma(((fma(fma(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0) * x) * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(fma(fma(0.37777777777777777, Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * x) * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.37777777777777777 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 14: 98.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* (* (fma 0.6666666666666666 (* x x) 1.0) x) x) eps eps))

C

double code(double x, double eps) {
	return fma(((fma(0.6666666666666666, (x * x), 1.0) * x) * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(fma(0.6666666666666666, Float64(x * x), 1.0) * x) * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 15: 98.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* x x) eps eps))

C

double code(double x, double eps) {
	return fma((x * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(x * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(x * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 64.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left({x}^{2}, \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(x \cdot x, \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Developer Target 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \left(\varepsilon \cdot \tan x\right) \cdot \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* (* eps (tan x)) (tan x))))

C

double code(double x, double eps) {
	return eps + ((eps * tan(x)) * tan(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((eps * tan(x)) * tan(x))
end function

Java

public static double code(double x, double eps) {
	return eps + ((eps * Math.tan(x)) * Math.tan(x));
}

Python

def code(x, eps):
	return eps + ((eps * math.tan(x)) * math.tan(x))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(eps * tan(x)) * tan(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((eps * tan(x)) * tan(x));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(eps * N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024340 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))