2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 9.2s

Alternatives: 12

Speedup: 13.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := {\tan x}^{3}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(1.3333333333333333, t\_0, \mathsf{fma}\left(\tan x, t\_1, 0.3333333333333333\right)\right), t\_1 + \tan x\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)) (t_1 (pow (tan x) 3.0)))
   (fma
    (fma
     (fma
      eps
      (fma 1.3333333333333333 t_0 (fma (tan x) t_1 0.3333333333333333))
      (+ t_1 (tan x)))
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = pow(tan(x), 3.0);
	return fma(fma(fma(eps, fma(1.3333333333333333, t_0, fma(tan(x), t_1, 0.3333333333333333)), (t_1 + tan(x))), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = tan(x) ^ 3.0
	return fma(fma(fma(eps, fma(1.3333333333333333, t_0, fma(tan(x), t_1, 0.3333333333333333)), Float64(t_1 + tan(x))), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]}, N[(N[(N[(eps * N[(1.3333333333333333 * t$95$0 + N[(N[Tan[x], $MachinePrecision] * t$95$1 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right) \]

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(1.3333333333333333, {\tan x}^{2}, \mathsf{fma}\left(\tan x, {\tan x}^{3}, 0.3333333333333333\right)\right), \tan x + {\tan x}^{3}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(1.3333333333333333, {\tan x}^{2}, \mathsf{fma}\left(\tan x, {\tan x}^{3}, 0.3333333333333333\right)\right), {\tan x}^{3} + \tan x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x \cdot \varepsilon, 0.16666666666666666, -0.5 \cdot \cos x\right) \cdot \varepsilon - \sin x, \varepsilon, \cos x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (*
   (fma
    (-
     (* (fma (* (sin x) eps) 0.16666666666666666 (* -0.5 (cos x))) eps)
     (sin x))
    eps
    (cos x))
   (cos x))))

C

double code(double x, double eps) {
	return sin(eps) / (fma(((fma((sin(x) * eps), 0.16666666666666666, (-0.5 * cos(x))) * eps) - sin(x)), eps, cos(x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(fma(Float64(Float64(fma(Float64(sin(x) * eps), 0.16666666666666666, Float64(-0.5 * cos(x))) * eps) - sin(x)), eps, cos(x)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision] * 0.16666666666666666 + N[(-0.5 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] * eps + N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

10. Applied rewrites 100.0%

    \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\sin x \cdot \varepsilon, 0.16666666666666666, -0.5 \cdot \cos x\right) \cdot \varepsilon - \sin x, \varepsilon, \cos x\right)} \cdot \cos x} \]
11. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (* (fma (- (* (* -0.5 (cos x)) eps) (sin x)) eps (cos x)) (cos x))))

C

double code(double x, double eps) {
	return sin(eps) / (fma((((-0.5 * cos(x)) * eps) - sin(x)), eps, cos(x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(fma(Float64(Float64(Float64(-0.5 * cos(x)) * eps) - sin(x)), eps, cos(x)) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower--.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-cos.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 5: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 6: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \varepsilon, 0\right), \varepsilon, 1\right), \varepsilon, 0\right)}{\cos \left(\varepsilon + x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (fma (fma (fma -0.16666666666666666 eps 0.0) eps 1.0) eps 0.0)
  (* (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	return fma(fma(fma(-0.16666666666666666, eps, 0.0), eps, 1.0), eps, 0.0) / (cos((eps + x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(-0.16666666666666666, eps, 0.0), eps, 1.0), eps, 0.0) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.16666666666666666 * eps + 0.0), $MachinePrecision] * eps + 1.0), $MachinePrecision] * eps + 0.0), $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

7. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \varepsilon, 0\right), \varepsilon, 1\right), \varepsilon, 0\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 7: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{0.5 \cdot \left(\cos \varepsilon + \cos \left(\left(\varepsilon + x\right) + x\right)\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ eps (* 0.5 (+ (cos eps) (cos (+ (+ eps x) x))))))

C

double code(double x, double eps) {
	return eps / (0.5 * (cos(eps) + cos(((eps + x) + x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (0.5d0 * (cos(eps) + cos(((eps + x) + x))))
end function

Java

public static double code(double x, double eps) {
	return eps / (0.5 * (Math.cos(eps) + Math.cos(((eps + x) + x))));
}

Python

def code(x, eps):
	return eps / (0.5 * (math.cos(eps) + math.cos(((eps + x) + x))))

Julia

function code(x, eps)
	return Float64(eps / Float64(0.5 * Float64(cos(eps) + cos(Float64(Float64(eps + x) + x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (0.5 * (cos(eps) + cos(((eps + x) + x))));
end

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. cos-sin-sum N/A

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

   9. *-rgt-identity N/A

      \[\leadsto \frac{0 + \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. lower-+.f64 99.7

       \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{0 + \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{0 + \varepsilon}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{0 + \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   4. lift-cos.f64 N/A

      \[\leadsto \frac{0 + \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\cos x}} \]

   5. cos-mult N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. associate--l+ N/A

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

   10. +-inverses N/A

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

   11. +-rgt-identity N/A

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

   12. lower-+.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. lower-cos.f64 99.7

       \[\leadsto \frac{0 + \varepsilon}{\left(\cos \left(\left(\varepsilon + x\right) + x\right) + \color{blue}{\cos \varepsilon}\right) \cdot 0.5} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{0 + \varepsilon}{\color{blue}{\left(\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \varepsilon\right) \cdot 0.5}} \]

10. Final simplification 99.7%

    \[\leadsto \frac{\varepsilon}{0.5 \cdot \left(\cos \varepsilon + \cos \left(\left(\varepsilon + x\right) + x\right)\right)} \]
11. Add Preprocessing

Alternative 8: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. cos-sin-sum N/A

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

   9. *-rgt-identity N/A

      \[\leadsto \frac{0 + \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. lower-+.f64 99.7

       \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

9. Applied rewrites 99.7%

   \[\leadsto \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
10. Add Preprocessing

Alternative 9: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (cos(x) ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps / Math.pow(Math.cos(x), 2.0);
}

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(eps / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-tan.f64 N/A

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

   4. tan-quot N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. distribute-neg-frac N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. cos-sin-sum N/A

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

   9. *-rgt-identity N/A

      \[\leadsto \frac{0 + \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   10. lower-+.f64 99.7

       \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{0 + \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{0 + \varepsilon}{\color{blue}{{\cos x}^{2}}} \]
9. Step-by-step derivation

   1. lower-pow.f64 N/A

      \[\leadsto \frac{0 + \varepsilon}{\color{blue}{{\cos x}^{2}}} \]

   2. lower-cos.f64 99.3

      \[\leadsto \frac{0 + \varepsilon}{{\color{blue}{\cos x}}^{2}} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{0 + \varepsilon}{\color{blue}{{\cos x}^{2}}} \]

11. Final simplification 99.3%

    \[\leadsto \frac{\varepsilon}{{\cos x}^{2}} \]
12. Add Preprocessing

Alternative 10: 98.3% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma (fma (* 1.3333333333333333 x) x 0.3333333333333333) eps x)
   eps
   (* x x))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(fma((1.3333333333333333 * x), x, 0.3333333333333333), eps, x), eps, (x * x)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(fma(Float64(1.3333333333333333 * x), x, 0.3333333333333333), eps, x), eps, Float64(x * x)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(1.3333333333333333 * x), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * eps + x), $MachinePrecision] * eps + N[(x * x), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right) \]

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.0%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333 \cdot x, x, 0.3333333333333333\right), \varepsilon, x\right), \varepsilon, x \cdot x\right), \varepsilon, \varepsilon\right) \]
15. Add Preprocessing

Alternative 11: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{1}{3} + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) - \left(\frac{-1}{2} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{1}{6} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon, \varepsilon\right) \]

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.0%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.9%

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

15. Final simplification 98.9%

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

Alternative 12: 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 63.0%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 63.0

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 63.0

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in eps around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 5.6

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

7. Applied rewrites 5.6%

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

Developer Target 1: 99.0% 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 2024332 
(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)))