2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 17.1s

Alternatives: 22

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: 62.5% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(-\tan \varepsilon, \tan x, 1\right)\right)}^{-1}, \tan \varepsilon + \tan x, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 7e-9)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (fma
    (pow (fma (- (tan eps)) (tan x) 1.0) -1.0)
    (+ (tan eps) (tan x))
    (- (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 7e-9) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = fma(pow(fma(-tan(eps), tan(x), 1.0), -1.0), (tan(eps) + tan(x)), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 7e-9)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = fma((fma(Float64(-tan(eps)), tan(x), 1.0) ^ -1.0), Float64(tan(eps) + tan(x)), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 7e-9], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[Power[N[((-N[Tan[eps], $MachinePrecision]) * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.9999999999999998e-9

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 6.9999999999999998e-9 < eps

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. tan-sum N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 88.5%

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

Alternative 2: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{{\sin x}^{2}}{t\_0}\\ t_2 := -0.16666666666666666 - \left(\mathsf{fma}\left(t\_1, 0.16666666666666666, \mathsf{fma}\left(t\_1, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{t\_0}\right)}{t\_0}\right)\\ t_3 := \frac{\frac{{\sin x}^{3}}{t\_0} + \sin x}{\cos x}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \frac{\sin x}{\cos x}, t\_3 \cdot 0.3333333333333333\right), \varepsilon, t\_2\right), \varepsilon, t\_3\right), \varepsilon, t\_1\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (/ (pow (sin x) 2.0) t_0))
        (t_2
         (-
          -0.16666666666666666
          (-
           (fma t_1 0.16666666666666666 (fma t_1 -0.5 -0.5))
           (/ (fma (sin x) (sin x) (/ (pow (sin x) 4.0) t_0)) t_0))))
        (t_3 (/ (+ (/ (pow (sin x) 3.0) t_0) (sin x)) (cos x))))
   (fma
    (fma
     (fma
      (fma (fma t_2 (/ (sin x) (cos x)) (* t_3 0.3333333333333333)) eps t_2)
      eps
      t_3)
     eps
     t_1)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0) / t_0;
	double t_2 = -0.16666666666666666 - (fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - (fma(sin(x), sin(x), (pow(sin(x), 4.0) / t_0)) / t_0));
	double t_3 = ((pow(sin(x), 3.0) / t_0) + sin(x)) / cos(x);
	return fma(fma(fma(fma(fma(t_2, (sin(x) / cos(x)), (t_3 * 0.3333333333333333)), eps, t_2), eps, t_3), eps, t_1), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64((sin(x) ^ 2.0) / t_0)
	t_2 = Float64(-0.16666666666666666 - Float64(fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - Float64(fma(sin(x), sin(x), Float64((sin(x) ^ 4.0) / t_0)) / t_0)))
	t_3 = Float64(Float64(Float64((sin(x) ^ 3.0) / t_0) + sin(x)) / cos(x))
	return fma(fma(fma(fma(fma(t_2, Float64(sin(x) / cos(x)), Float64(t_3 * 0.3333333333333333)), eps, t_2), eps, t_3), eps, t_1), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.16666666666666666 - N[(N[(t$95$1 * 0.16666666666666666 + N[(t$95$1 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + eps), $MachinePrecision]]]]]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x} \cdot 0.3333333333333333\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right) \]
7. Add Preprocessing

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := {\cos x}^{-2}\\ t_2 := \mathsf{fma}\left(t\_1, \mathsf{fma}\left(t\_1, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left(t\_0, -0.3333333333333333, -0.5\right)\right)\\ t_3 := \frac{\mathsf{fma}\left(\sin x, t\_0, \sin x\right)}{\cos x}\\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, t\_2, t\_3 \cdot 0.3333333333333333\right), t\_2\right), \varepsilon, t\_3\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (pow (cos x) -2.0))
        (t_2
         (fma
          t_1
          (fma t_1 (pow (sin x) 4.0) (pow (sin x) 2.0))
          (- -0.16666666666666666 (fma t_0 -0.3333333333333333 -0.5))))
        (t_3 (/ (fma (sin x) t_0 (sin x)) (cos x))))
   (fma
    (fma
     (tan x)
     (tan x)
     (*
      (fma (fma eps (fma (tan x) t_2 (* t_3 0.3333333333333333)) t_2) eps t_3)
      eps))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = pow(cos(x), -2.0);
	double t_2 = fma(t_1, fma(t_1, pow(sin(x), 4.0), pow(sin(x), 2.0)), (-0.16666666666666666 - fma(t_0, -0.3333333333333333, -0.5)));
	double t_3 = fma(sin(x), t_0, sin(x)) / cos(x);
	return fma(fma(tan(x), tan(x), (fma(fma(eps, fma(tan(x), t_2, (t_3 * 0.3333333333333333)), t_2), eps, t_3) * eps)), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = cos(x) ^ -2.0
	t_2 = fma(t_1, fma(t_1, (sin(x) ^ 4.0), (sin(x) ^ 2.0)), Float64(-0.16666666666666666 - fma(t_0, -0.3333333333333333, -0.5)))
	t_3 = Float64(fma(sin(x), t_0, sin(x)) / cos(x))
	return fma(fma(tan(x), tan(x), Float64(fma(fma(eps, fma(tan(x), t_2, Float64(t_3 * 0.3333333333333333)), t_2), eps, t_3) * eps)), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 * N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 - N[(t$95$0 * -0.3333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$0 + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(N[(N[(eps * N[(N[Tan[x], $MachinePrecision] * t$95$2 + N[(t$95$3 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]]]]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]
8. Add Preprocessing

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\\ t_2 := \mathsf{fma}\left(t\_0, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, t\_1, \mathsf{fma}\left(t\_2, \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, t\_2\right)\right), \varepsilon, t\_1\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (* (fma (tan x) (tan x) 1.0) (tan x)))
        (t_2
         (+
          (fma t_0 0.3333333333333333 0.3333333333333333)
          (pow (tan x) 4.0))))
   (fma
    (fma
     (fma
      (fma
       eps
       (fma 0.3333333333333333 t_1 (fma t_2 (tan x) (pow (tan x) 3.0)))
       (fma (tan x) (tan x) t_2))
      eps
      t_1)
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = fma(tan(x), tan(x), 1.0) * tan(x);
	double t_2 = fma(t_0, 0.3333333333333333, 0.3333333333333333) + pow(tan(x), 4.0);
	return fma(fma(fma(fma(eps, fma(0.3333333333333333, t_1, fma(t_2, tan(x), pow(tan(x), 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(fma(tan(x), tan(x), 1.0) * tan(x))
	t_2 = Float64(fma(t_0, 0.3333333333333333, 0.3333333333333333) + (tan(x) ^ 4.0))
	return fma(fma(fma(fma(eps, fma(0.3333333333333333, t_1, fma(t_2, tan(x), (tan(x) ^ 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * 0.3333333333333333 + 0.3333333333333333), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(eps * N[(0.3333333333333333 * t$95$1 + N[(t$95$2 * N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]]]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

9. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x, \mathsf{fma}\left({\tan x}^{4} + \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right), \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right)\right)\right), \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \color{blue}{\varepsilon}, \varepsilon\right) \]

12. Final simplification 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x, \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}, \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}\right)\right), \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 5: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{t\_0}, 0.3333333333333333, 0.3333333333333333\right) + \mathsf{fma}\left(\sin x, \frac{\sin x}{t\_0}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)))
   (fma
    (fma
     (tan x)
     (tan x)
     (*
      (fma
       (+
        (fma (/ (pow (sin x) 2.0) t_0) 0.3333333333333333 0.3333333333333333)
        (fma (sin x) (/ (sin x) t_0) (/ (pow (sin x) 4.0) (pow (cos x) 4.0))))
       eps
       (/ (fma (sin x) (pow (tan x) 2.0) (sin x)) (cos x)))
      eps))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	return fma(fma(tan(x), tan(x), (fma((fma((pow(sin(x), 2.0) / t_0), 0.3333333333333333, 0.3333333333333333) + fma(sin(x), (sin(x) / t_0), (pow(sin(x), 4.0) / pow(cos(x), 4.0)))), eps, (fma(sin(x), pow(tan(x), 2.0), sin(x)) / cos(x))) * eps)), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	return fma(fma(tan(x), tan(x), Float64(fma(Float64(fma(Float64((sin(x) ^ 2.0) / t_0), 0.3333333333333333, 0.3333333333333333) + fma(sin(x), Float64(sin(x) / t_0), Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))), eps, Float64(fma(sin(x), (tan(x) ^ 2.0), sin(x)) / cos(x))) * eps)), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(N[(N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] * 0.3333333333333333 + 0.3333333333333333), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[Sin[x], $MachinePrecision] * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

8. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \frac{-1}{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right) + \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.3333333333333333, 0.3333333333333333\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

11. Final simplification 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.3333333333333333, 0.3333333333333333\right) + \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, \frac{{\sin x}^{4}}{{\cos x}^{4}}\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 6: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot t\_0\right) + {\tan x}^{4}\right) + t\_0\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (fma
    (fma
     (fma
      (fma
       (* 0.6666666666666666 x)
       eps
       (+
        (+
         (- 0.3333333333333333 (* -0.3333333333333333 t_0))
         (pow (tan x) 4.0))
        t_0))
      eps
      (/ (* (fma (tan x) (tan x) 1.0) (sin x)) (cos x)))
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return fma(fma(fma(fma((0.6666666666666666 * x), eps, (((0.3333333333333333 - (-0.3333333333333333 * t_0)) + pow(tan(x), 4.0)) + t_0)), eps, ((fma(tan(x), tan(x), 1.0) * sin(x)) / cos(x))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return fma(fma(fma(fma(Float64(0.6666666666666666 * x), eps, Float64(Float64(Float64(0.3333333333333333 - Float64(-0.3333333333333333 * t_0)) + (tan(x) ^ 4.0)) + t_0)), eps, Float64(Float64(fma(tan(x), tan(x), 1.0) * sin(x)) / cos(x))), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * eps + N[(N[(N[(0.3333333333333333 - N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

9. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, \varepsilon, \left(\left(\frac{1}{3} - \frac{-1}{3} \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

12. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 7: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    0.3333333333333333
    eps
    (/ (* (fma (tan x) (tan x) 1.0) (sin x)) (cos x)))
   eps
   (pow (tan x) 2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(0.3333333333333333, eps, ((fma(tan(x), tan(x), 1.0) * sin(x)) / cos(x))), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(0.3333333333333333, eps, Float64(Float64(fma(tan(x), tan(x), 1.0) * sin(x)) / cos(x))), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

9. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 8: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{\mathsf{fma}\left(-\tan \varepsilon, \tan x, 1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-9)
   (fma (pow (tan x) 2.0) eps eps)
   (- (/ (+ (tan eps) (tan x)) (fma (- (tan eps)) (tan x) 1.0)) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-9) {
		tmp = fma(pow(tan(x), 2.0), eps, eps);
	} else {
		tmp = ((tan(eps) + tan(x)) / fma(-tan(eps), tan(x), 1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-9)
		tmp = fma((tan(x) ^ 2.0), eps, eps);
	else
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / fma(Float64(-tan(eps)), tan(x), 1.0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-9], N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[eps], $MachinePrecision]) * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000006e-9

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if 1.00000000000000006e-9 < eps

   1. Initial program 78.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-tan.f64 N/A

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

      12. *-commutative N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-tan.f64 85.8

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

   4. Applied rewrites 85.8%

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

Alternative 9: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right) \cdot x\right)\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (tan x)
   (tan x)
   (fma
    (* 0.3333333333333333 eps)
    eps
    (* (fma 0.6666666666666666 (pow eps 3.0) eps) x)))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(tan(x), tan(x), fma((0.3333333333333333 * eps), eps, (fma(0.6666666666666666, pow(eps, 3.0), eps) * x))), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(tan(x), tan(x), fma(Float64(0.3333333333333333 * eps), eps, Float64(fma(0.6666666666666666, (eps ^ 3.0), eps) * x))), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + N[(N[(0.3333333333333333 * eps), $MachinePrecision] * eps + N[(N[(0.6666666666666666 * N[Power[eps, 3.0], $MachinePrecision] + eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\tan x, \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x} \cdot 0.3333333333333333\right), \mathsf{fma}\left({\cos x}^{-2}, \mathsf{fma}\left({\cos x}^{-2}, {\sin x}^{4}, {\sin x}^{2}\right), -0.16666666666666666 - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right)\right), \varepsilon, \frac{\mathsf{fma}\left(\sin x, {\tan x}^{2}, \sin x\right)}{\cos x}\right) \cdot \varepsilon\right), \varepsilon, \varepsilon\right) \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(\varepsilon \cdot 0.3333333333333333, \varepsilon, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right) \cdot x\right)\right), \varepsilon, \varepsilon\right) \]

11. Final simplification 97.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left(0.3333333333333333 \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right) \cdot x\right)\right), \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 10: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(\varepsilon + x\right) - x\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.35e-8)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (/ (sin (- (+ eps x) x)) (* (cos (+ eps x)) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.35e-8) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = sin(((eps + x) - x)) / (cos((eps + x)) * cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.35e-8)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = Float64(sin(Float64(Float64(eps + x) - x)) / Float64(cos(Float64(eps + x)) * cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.35e-8], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[Sin[N[(N[(eps + x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.35000000000000001e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1.35000000000000001e-8 < eps

   1. Initial program 81.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. tan-quot N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sin-diff N/A

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

      9. lower-sin.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lower-cos.f64 82.3

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

   4. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\frac{\sin \left(\left(\varepsilon + x\right) - x\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \tan \left(\varepsilon + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-8)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (fma (/ -1.0 (cos x)) (sin x) (tan (+ eps x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-8) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = fma((-1.0 / cos(x)), sin(x), tan((eps + x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-8)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = fma(Float64(-1.0 / cos(x)), sin(x), tan(Float64(eps + x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-8], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1e-8 < eps

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-cos.f64 81.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 81.6

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 81.6

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

   6. Applied rewrites 81.6%

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

Alternative 12: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \left(\varepsilon + x\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.35e-8)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (- (/ (sin (+ eps x)) (cos (+ eps x))) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.35e-8) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = (sin((eps + x)) / cos((eps + x))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.35e-8)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = Float64(Float64(sin(Float64(eps + x)) / cos(Float64(eps + x))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.35e-8], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.35000000000000001e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1.35000000000000001e-8 < eps

   1. Initial program 81.3%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 81.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 81.4

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

   4. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{\sin \left(\varepsilon + x\right)}{\cos \left(\varepsilon + x\right)}} - \tan x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \frac{\sin x}{\cos x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-8)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (- (tan (+ eps x)) (/ (sin x) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-8) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = tan((eps + x)) - (sin(x) / cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-8)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = Float64(tan(Float64(eps + x)) - Float64(sin(x) / cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-8], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1e-8 < eps

   1. Initial program 81.3%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 81.4

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

   4. Applied rewrites 81.4%

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \frac{\sin x}{\cos x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-8)
   (fma (/ 1.0 (pow (tan x) -2.0)) eps eps)
   (- (tan (+ eps x)) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-8) {
		tmp = fma((1.0 / pow(tan(x), -2.0)), eps, eps);
	} else {
		tmp = tan((eps + x)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-8)
		tmp = fma(Float64(1.0 / (tan(x) ^ -2.0)), eps, eps);
	else
		tmp = Float64(tan(Float64(eps + x)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-8], N[(N[(1.0 / N[Power[N[Tan[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1e-8 < eps

   1. Initial program 81.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{{\tan x}^{-2}}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-8)
   (fma (pow (tan x) 2.0) eps eps)
   (- (tan (+ eps x)) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-8) {
		tmp = fma(pow(tan(x), 2.0), eps, eps);
	} else {
		tmp = tan((eps + x)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-8)
		tmp = fma((tan(x) ^ 2.0), eps, eps);
	else
		tmp = Float64(tan(Float64(eps + x)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-8], N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1e-8

   1. Initial program 61.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   if 1e-8 < eps

   1. Initial program 81.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.44 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.44e-11)
   (fma
    (*
     (*
      (fma
       (fma
        (fma 0.19682539682539682 (* x x) 0.37777777777777777)
        (* x x)
        0.6666666666666666)
       (* x x)
       1.0)
      x)
     x)
    eps
    eps)
   (- (tan (+ eps x)) (tan x))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.44e-11) {
		tmp = fma(((fma(fma(fma(0.19682539682539682, (x * x), 0.37777777777777777), (x * x), 0.6666666666666666), (x * x), 1.0) * x) * x), eps, eps);
	} else {
		tmp = tan((eps + x)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.44e-11)
		tmp = 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);
	else
		tmp = Float64(tan(Float64(eps + x)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.44e-11], 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], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.44e-11

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.7%

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

   if 1.44e-11 < eps

   1. Initial program 75.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.44 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.19682539682539682, x \cdot x, 0.37777777777777777\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot x, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right), x, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    (fma
     (* (fma 1.8888888888888888 (* eps eps) 1.3333333333333333) x)
     eps
     (fma 1.3333333333333333 (* eps eps) 1.0))
    x
    (fma 0.6666666666666666 (pow eps 3.0) eps))
   x
   (* (* eps eps) 0.3333333333333333))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma((fma(1.8888888888888888, (eps * eps), 1.3333333333333333) * x), eps, fma(1.3333333333333333, (eps * eps), 1.0)), x, fma(0.6666666666666666, pow(eps, 3.0), eps)), x, ((eps * eps) * 0.3333333333333333)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(Float64(fma(1.8888888888888888, Float64(eps * eps), 1.3333333333333333) * x), eps, fma(1.3333333333333333, Float64(eps * eps), 1.0)), x, fma(0.6666666666666666, (eps ^ 3.0), eps)), x, Float64(Float64(eps * eps) * 0.3333333333333333)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(1.8888888888888888 * N[(eps * eps), $MachinePrecision] + 1.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * eps + N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(0.6666666666666666 * N[Power[eps, 3.0], $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{1}{6} \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(\frac{1}{6} + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(\frac{-1}{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right), x, \mathsf{fma}\left(0.6666666666666666, {\varepsilon}^{3}, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 18: 98.1% 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 61.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg 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 97.3

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.2%

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

Alternative 19: 98.1% 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 61.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg 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 97.3

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.1%

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

Alternative 20: 98.1% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg 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 97.3

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.0%

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

Alternative 21: 98.0% 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 61.9%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg 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 97.3

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.9%

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

Alternative 22: 5.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 61.9

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 61.9

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in eps around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{\sin x}{\cos x} \]

   3. mul0-lft 5.3

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 5.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.8% 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 2024259 
(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)))