2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.6%

Time: 15.7s

Alternatives: 15

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \cos \left(x + x\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_1, -0.5, 0.5\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{\mathsf{fma}\left(t\_1, 0.5, 0.5\right)} - \mathsf{fma}\left(t\_0, -0.3333333333333333, -0.3333333333333333\right), \varepsilon, \tan x \cdot \left(t\_0 + 1\right)\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (cos (+ x x))))
   (fma
    (fma
     (fma
      (-
       (/
        (+ (fma t_1 -0.5 0.5) (pow (* (sin x) (tan x)) 2.0))
        (fma t_1 0.5 0.5))
       (fma t_0 -0.3333333333333333 -0.3333333333333333))
      eps
      (* (tan x) (+ t_0 1.0)))
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = cos((x + x));
	return fma(fma(fma((((fma(t_1, -0.5, 0.5) + pow((sin(x) * tan(x)), 2.0)) / fma(t_1, 0.5, 0.5)) - fma(t_0, -0.3333333333333333, -0.3333333333333333)), eps, (tan(x) * (t_0 + 1.0))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = cos(Float64(x + x))
	return fma(fma(fma(Float64(Float64(Float64(fma(t_1, -0.5, 0.5) + (Float64(sin(x) * tan(x)) ^ 2.0)) / fma(t_1, 0.5, 0.5)) - fma(t_0, -0.3333333333333333, -0.3333333333333333)), eps, Float64(tan(x) * Float64(t_0 + 1.0))), 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[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] + N[Power[N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * -0.3333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + N[(N[Tan[x], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)} - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.3333333333333333\right), \frac{\left({\tan x}^{2} + 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right)}, \varepsilon, \varepsilon\right) \]

11. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{\mathsf{fma}\left(\cos \left(x + x\right), 0.5, 0.5\right)} - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.3333333333333333\right), \varepsilon, \tan x \cdot \left({\tan x}^{2} + 1\right)\right)}, \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, t\_0 + {t\_0}^{3}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (fma (fma eps (+ t_0 (pow t_0 3.0)) (pow (tan x) 2.0)) eps eps)))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return fma(fma(eps, (t_0 + pow(t_0, 3.0)), pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return fma(fma(eps, Float64(t_0 + (t_0 ^ 3.0)), (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in eps around 0

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. unpow2 N/A

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

   11. unpow2 N/A

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

   12. times-frac N/A

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

   13. cube-unmult N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 98.6

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

10. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (fma
    (fma
     (fma eps 0.3333333333333333 (/ (* (sin x) (+ t_0 1.0)) (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(eps, 0.3333333333333333, ((sin(x) * (t_0 + 1.0)) / cos(x))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return fma(fma(fma(eps, 0.3333333333333333, Float64(Float64(sin(x) * Float64(t_0 + 1.0)) / 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[(eps * 0.3333333333333333 + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{\mathsf{fma}\left(\cos \left(x + x\right), -0.5, 0.5\right) + {\left(\sin x \cdot \tan x\right)}^{2}}{\mathsf{fma}\left(0.5, \cos \left(x + x\right), 0.5\right)} - \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.3333333333333333\right), \frac{\left({\tan x}^{2} + 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right)}, \varepsilon, \varepsilon\right) \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.6%

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

13. Final simplification 98.6%

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

Alternative 4: 99.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma eps (fma eps 0.3333333333333333 x) (pow (tan x) 2.0)) eps eps))

C

double code(double x, double eps) {
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(eps, fma(eps, 0.3333333333333333, x), (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 98.2

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

10. Applied rewrites 98.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, 0.3333333333333333, x\right)}, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 98.1

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

7. Applied rewrites 98.1%

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

Alternative 6: 98.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}, \varepsilon\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right) + 1\right)}, \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right)}, \varepsilon\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right), \varepsilon\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right), \varepsilon\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right) + \frac{2}{3}}, 1\right), \varepsilon\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right)}, 1\right), \varepsilon\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{62}{315} \cdot {x}^{2} + \frac{17}{45}}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{62}{315}} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{62}{315} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{62}{315}\right)} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right)}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   17. lower-*.f64 97.8

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

8. Applied rewrites 97.8%

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

Alternative 7: 98.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right)}, \varepsilon\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right) + 1\right)}, \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right)}, \varepsilon\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right), \varepsilon\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right), 1\right), \varepsilon\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right) + \frac{2}{3}}, 1\right), \varepsilon\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right)}, 1\right), \varepsilon\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45} + \frac{62}{315} \cdot {x}^{2}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{62}{315} \cdot {x}^{2} + \frac{17}{45}}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{62}{315}} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{62}{315} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{62}{315}\right)} + \frac{17}{45}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right)}, \frac{2}{3}\right), 1\right), \varepsilon\right) \]

   17. lower-*.f64 97.8

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

8. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right)}, \varepsilon\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{62}{315}\right) + \frac{17}{45}\right) + \frac{2}{3}\right) + 1\right)\right) + \varepsilon \]

   2. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{62}{315}\right) + \frac{17}{45}\right) + \frac{2}{3}\right) + 1\right)\right) + \varepsilon \]

   3. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{62}{315}\right) + \frac{17}{45}\right) + \frac{2}{3}\right) + 1\right)\right) + \varepsilon \]

   4. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{62}{315}\right)} + \frac{17}{45}\right) + \frac{2}{3}\right) + 1\right)\right) + \varepsilon \]

   5. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right)} + \frac{2}{3}\right) + 1\right)\right) + \varepsilon \]

   6. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right), \frac{2}{3}\right)} + 1\right)\right) + \varepsilon \]

   7. lift-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right), \frac{2}{3}\right), 1\right)}\right) + \varepsilon \]

   8. lift-*.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{62}{315}, \frac{17}{45}\right), \frac{2}{3}\right), 1\right)\right)} + \varepsilon \]

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 8: 98.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right), \varepsilon\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}, \varepsilon\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}, \varepsilon\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}, \varepsilon\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right)}\right), \varepsilon\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)}\right), \varepsilon\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)\right), \varepsilon\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)\right), \varepsilon\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}}, 1\right)\right), \varepsilon\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{17}{45}} + \frac{2}{3}, 1\right)\right), \varepsilon\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45}, \frac{2}{3}\right)}, 1\right)\right), \varepsilon\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45}, \frac{2}{3}\right), 1\right)\right), \varepsilon\right) \]

   13. lower-*.f64 97.7

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

8. Applied rewrites 97.7%

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

Alternative 9: 98.2% accurate, 5.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (*
   x
   (* x (fma (* x x) (fma 0.3333333333333333 (* x x) 0.6666666666666666) 1.0)))
  eps))

C

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

Julia

function code(x, eps)
	return fma(eps, Float64(x * Float64(x * fma(Float64(x * x), fma(0.3333333333333333, Float64(x * x), 0.6666666666666666), 1.0))), eps)
end

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 97.6

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 97.7

        \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.3333333333333333, \color{blue}{x \cdot x}, 0.6666666666666666\right), 1\right)\right), \varepsilon\right) \]

11. Applied rewrites 97.7%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.3333333333333333, x \cdot x, 0.6666666666666666\right), 1\right)\right)}, \varepsilon\right) \]
12. Add Preprocessing

Alternative 10: 98.2% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. distribute-rgt-out-- N/A

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot \color{blue}{\frac{2}{3}}, \varepsilon\right), \varepsilon\right) \]

   11. lower-*.f64 97.6

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

8. Applied rewrites 97.6%

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

Alternative 11: 98.2% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma eps (fma x (+ x eps) (* 0.3333333333333333 (* eps eps))) eps))

C

double code(double x, double eps) {
	return fma(eps, fma(x, (x + eps), (0.3333333333333333 * (eps * eps))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(x, Float64(x + eps), Float64(0.3333333333333333 * Float64(eps * eps))), eps)
end

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{4}{3}, 1\right), \varepsilon\right), \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

   12. lower-*.f64 97.4

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right), 0.3333333333333333 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

8. Applied rewrites 97.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 97.4

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x + \varepsilon}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

11. Applied rewrites 97.4%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x + \varepsilon}, 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]
12. Add Preprocessing

Alternative 12: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{4}{3}, 1\right), \varepsilon\right), \frac{1}{3} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

   12. lower-*.f64 97.4

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right), 0.3333333333333333 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), \varepsilon\right) \]

8. Applied rewrites 97.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), \varepsilon\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. distribute-rgt-out N/A

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

    3. lower-*.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-+.f64 97.4

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

11. Applied rewrites 97.4%

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

Alternative 13: 98.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Applied rewrites 97.4%

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

Alternative 14: 98.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Applied rewrites 97.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot x}, \varepsilon\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 97.4

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

10. Applied rewrites 97.4%

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

11. Final simplification 97.4%

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

Alternative 15: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x * (x * eps)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 97.4

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

8. Applied rewrites 97.4%

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

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\varepsilon \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{{x}^{2} \cdot \varepsilon} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \varepsilon \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \varepsilon\right)} \]

    4. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(\varepsilon \cdot x\right)} \]

    5. lower-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(\varepsilon \cdot x\right)} \]

    6. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \varepsilon\right)} \]

    7. lower-*.f64 6.5

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \varepsilon\right)} \]

11. Applied rewrites 6.5%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot \varepsilon\right)} \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

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

Developer Target 2: 62.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% 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 2024216 
(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 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

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

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