2tan (problem 3.3.2)

Percentage Accurate: 62.8% → 99.7%

Time: 16.2s

Alternatives: 17

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: 62.8% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := -t\_0\\ t_2 := \mathsf{fma}\left(\tan x, \tan x, 1\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t\_2 \cdot \mathsf{fma}\left(\sin x, -0.6666666666666666, t\_1 \cdot \sin x\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, t\_0, \mathsf{fma}\left(t\_1, t\_2, \mathsf{fma}\left(t\_0, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, t\_0, \varepsilon\right), \sin x, \sin x \cdot \tan x\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (- t_0)) (t_2 (fma (tan x) (tan x) 1.0)))
   (fma
    (fma
     (*
      (fma
       (/ (* t_2 (fma (sin x) -0.6666666666666666 (* t_1 (sin x)))) (cos x))
       (- eps)
       (-
        -0.16666666666666666
        (fma 0.16666666666666666 t_0 (fma t_1 t_2 (fma t_0 -0.5 -0.5)))))
      eps)
     eps
     (/ (fma (fma eps t_0 eps) (sin x) (* (sin x) (tan x))) (cos x)))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = -t_0;
	double t_2 = fma(tan(x), tan(x), 1.0);
	return fma(fma((fma(((t_2 * fma(sin(x), -0.6666666666666666, (t_1 * sin(x)))) / cos(x)), -eps, (-0.16666666666666666 - fma(0.16666666666666666, t_0, fma(t_1, t_2, fma(t_0, -0.5, -0.5))))) * eps), eps, (fma(fma(eps, t_0, eps), sin(x), (sin(x) * tan(x))) / cos(x))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(-t_0)
	t_2 = fma(tan(x), tan(x), 1.0)
	return fma(fma(Float64(fma(Float64(Float64(t_2 * fma(sin(x), -0.6666666666666666, Float64(t_1 * sin(x)))) / cos(x)), Float64(-eps), Float64(-0.16666666666666666 - fma(0.16666666666666666, t_0, fma(t_1, t_2, fma(t_0, -0.5, -0.5))))) * eps), eps, Float64(fma(fma(eps, t_0, eps), sin(x), Float64(sin(x) * tan(x))) / cos(x))), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] * -0.6666666666666666 + N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * (-eps) + N[(-0.16666666666666666 - N[(0.16666666666666666 * t$95$0 + N[(t$95$1 * t$95$2 + N[(t$95$0 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(eps * t$95$0 + eps), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]]]

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right), -0.3333333333333333, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right) \cdot \frac{-{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right) \cdot -0.3333333333333333\right)\right)}{\cos x}, -0.16666666666666666 - \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right), -0.5, \frac{\mathsf{fma}\left({\sin x}^{2}, 0.16666666666666666, \left(-{\sin x}^{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right)\right)}{{\cos x}^{2}}\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x \cdot \left(-{\tan x}^{2}\right), \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right), \sin x \cdot -0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \left(\sin x \cdot -0.3333333333333333\right)\right)\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\sin x, -0.6666666666666666, \left(-{\tan x}^{2}\right) \cdot \sin x\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\sin x, -0.6666666666666666, \left(-{\tan x}^{2}\right) \cdot \sin x\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right), \sin x, \sin x \cdot \tan x\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (pow (sin x) 2.0)) (t_2 (/ t_1 t_0)))
   (fma
    (fma
     (*
      (-
       0.3333333333333333
       (/ (fma -0.3333333333333333 t_1 (* (+ t_2 1.0) (- t_1))) t_0))
      eps)
     eps
     (/ (fma (fma t_2 eps eps) (sin x) (/ t_1 (cos x))) (cos x)))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = t_1 / t_0;
	return fma(fma(((0.3333333333333333 - (fma(-0.3333333333333333, t_1, ((t_2 + 1.0) * -t_1)) / t_0)) * eps), eps, (fma(fma(t_2, eps, eps), sin(x), (t_1 / cos(x))) / cos(x))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	return fma(fma(Float64(Float64(0.3333333333333333 - Float64(fma(-0.3333333333333333, t_1, Float64(Float64(t_2 + 1.0) * Float64(-t_1))) / t_0)) * eps), eps, Float64(fma(fma(t_2, eps, eps), sin(x), Float64(t_1 / cos(x))) / cos(x))), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right), -0.3333333333333333, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right) \cdot \frac{-{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right) \cdot -0.3333333333333333\right)\right)}{\cos x}, -0.16666666666666666 - \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right), -0.5, \frac{\mathsf{fma}\left({\sin x}^{2}, 0.16666666666666666, \left(-{\sin x}^{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right)\right)}{{\cos x}^{2}}\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x \cdot \left(-{\tan x}^{2}\right), \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right), \sin x \cdot -0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \left(\sin x \cdot -0.3333333333333333\right)\right)\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := {\sin x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.6666666666666666 \cdot x}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, t\_0, \mathsf{fma}\left(-t\_0, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(t\_0, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{t\_1}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{t\_1}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (pow (sin x) 2.0)))
   (fma
    (fma
     (*
      (fma
       (/ (* -0.6666666666666666 x) (cos x))
       (- eps)
       (-
        -0.16666666666666666
        (fma
         0.16666666666666666
         t_0
         (fma (- t_0) (fma (tan x) (tan x) 1.0) (fma t_0 -0.5 -0.5)))))
      eps)
     eps
     (/
      (fma (fma (/ t_1 (pow (cos x) 2.0)) eps eps) (sin x) (/ t_1 (cos x)))
      (cos x)))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	return fma(fma((fma(((-0.6666666666666666 * x) / cos(x)), -eps, (-0.16666666666666666 - fma(0.16666666666666666, t_0, fma(-t_0, fma(tan(x), tan(x), 1.0), fma(t_0, -0.5, -0.5))))) * eps), eps, (fma(fma((t_1 / pow(cos(x), 2.0)), eps, eps), sin(x), (t_1 / cos(x))) / cos(x))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	return fma(fma(Float64(fma(Float64(Float64(-0.6666666666666666 * x) / cos(x)), Float64(-eps), Float64(-0.16666666666666666 - fma(0.16666666666666666, t_0, fma(Float64(-t_0), fma(tan(x), tan(x), 1.0), fma(t_0, -0.5, -0.5))))) * eps), eps, Float64(fma(fma(Float64(t_1 / (cos(x) ^ 2.0)), eps, eps), sin(x), Float64(t_1 / cos(x))) / cos(x))), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[(N[(-0.6666666666666666 * x), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * (-eps) + N[(-0.16666666666666666 - N[(0.16666666666666666 * t$95$0 + N[((-t$95$0) * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] + N[(t$95$0 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(N[(t$95$1 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(t$95$1 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right), -0.3333333333333333, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right) \cdot \frac{-{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right) \cdot -0.3333333333333333\right)\right)}{\cos x}, -0.16666666666666666 - \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right), -0.5, \frac{\mathsf{fma}\left({\sin x}^{2}, 0.16666666666666666, \left(-{\sin x}^{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right)\right)}{{\cos x}^{2}}\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right)} \]

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x \cdot \left(-{\tan x}^{2}\right), \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right), \sin x \cdot -0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \left(\sin x \cdot -0.3333333333333333\right)\right)\right)}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.6666666666666666 \cdot x}{\cos x}, -\varepsilon, -0.16666666666666666 - \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left(-{\tan x}^{2}, \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-\varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right), -0.3333333333333333, \mathsf{fma}\left(\sin x, \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right) \cdot \frac{-{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \sin x, \sin x\right) \cdot -0.3333333333333333\right)\right)}{\cos x}, -0.16666666666666666 - \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right), -0.5, \frac{\mathsf{fma}\left({\sin x}^{2}, 0.16666666666666666, \left(-{\sin x}^{2}\right) \cdot \mathsf{fma}\left(\sin x, \frac{\sin x}{{\cos x}^{2}}, 1\right)\right)}{{\cos x}^{2}}\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right), \sin x, \frac{{\sin x}^{2}}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.6%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 6: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return ((math.pow(math.sin(x), 2.0) * eps) / math.pow(math.cos(x), 2.0)) + (math.tan(eps) / (1.0 - (math.tan(eps) * math.tan(x))))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (((sin(x) ^ 2.0) * eps) / (cos(x) ^ 2.0)) + (tan(eps) / (1.0 - (tan(eps) * tan(x))));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. tan-sum N/A

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

   7. lift-tan.f64 N/A

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

   8. div-add N/A

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

   9. associate-+r+ N/A

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

   10. lower-+.f64 N/A

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in eps around 0

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

   1. associate-+r+ N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} \cdot \frac{\sin x}{\cos x} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   4. mul0-lft N/A

      \[\leadsto \left(\color{blue}{0} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   5. sin-PI N/A

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

   6. lower-+.f64 N/A

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

   7. sin-PI N/A

      \[\leadsto \left(\color{blue}{0} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(0 + \color{blue}{\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   9. *-commutative N/A

      \[\leadsto \left(0 + \frac{\color{blue}{{\sin x}^{2} \cdot \varepsilon}}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(0 + \frac{\color{blue}{{\sin x}^{2} \cdot \varepsilon}}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   11. lower-pow.f64 N/A

       \[\leadsto \left(0 + \frac{\color{blue}{{\sin x}^{2}} \cdot \varepsilon}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   12. lower-sin.f64 N/A

       \[\leadsto \left(0 + \frac{{\color{blue}{\sin x}}^{2} \cdot \varepsilon}{{\cos x}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   13. lower-pow.f64 N/A

       \[\leadsto \left(0 + \frac{{\sin x}^{2} \cdot \varepsilon}{\color{blue}{{\cos x}^{2}}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

   14. lower-cos.f64 98.9

       \[\leadsto \left(0 + \frac{{\sin x}^{2} \cdot \varepsilon}{{\color{blue}{\cos x}}^{2}}\right) + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\left(0 + \frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}}\right)} + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]

8. Final simplification 98.9%

   \[\leadsto \frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} \]
9. Add Preprocessing

Alternative 7: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, x\right)\\ \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{\frac{\sin \left(t\_0 - x\right) + \sin \left(t\_0 + x\right)}{2}}, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (PI) 0.5 x)))
   (fma
    eps
    (/ (pow (sin x) 2.0) (/ (+ (sin (- t_0 x)) (sin (+ t_0 x))) 2.0))
    eps)))

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 8: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 9: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 10: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 11: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 12: 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 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 13: 98.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 14: 98.2% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 15: 98.1% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 16: 98.0% 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 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. clear-num N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. clear-num N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 38.0%

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

5. Taylor expanded in eps around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate-/l* N/A

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

   7. +-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-cos.f64 98.7

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

11. Final simplification 97.7%

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

Alternative 17: 5.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 63.7%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. lower-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-PI.f64 7.6

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

4. Applied rewrites 7.6%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\sin x}{\cos x} - \frac{\sin \left(x + \mathsf{PI}\left(\right)\right)}{\cos \left(x + \mathsf{PI}\left(\right)\right)}} \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. neg-mul-1 N/A

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

   3. sin-+PI N/A

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

   4. distribute-frac-neg N/A

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

   5. cos-+PI N/A

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

   6. distribute-neg-frac2 N/A

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

   7. remove-double-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-lft1-in N/A

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

   10. metadata-eval N/A

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

   11. mul0-lft 5.3

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

7. Applied rewrites 5.3%

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

8. Final simplification 5.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Developer Target 1: 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 2024305 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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