2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.6%

Time: 15.7s

Alternatives: 14

Speedup: 34.5×

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: 62.7% 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, 0.1× speedup

Math

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

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
    eps
    (fma
     eps
     (fma
      eps
      (+
       (-
        (/ (+ t_1 (/ (pow (sin x) 4.0) t_0)) t_0)
        (fma 0.16666666666666666 t_2 (fma -0.5 t_2 -0.5)))
       -0.16666666666666666)
      (/ (+ (sin x) (/ (pow (sin x) 3.0) t_0)) (cos x)))
     t_2)
    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(eps, fma(eps, fma(eps, ((((t_1 + (pow(sin(x), 4.0) / t_0)) / t_0) - fma(0.16666666666666666, t_2, fma(-0.5, t_2, -0.5))) + -0.16666666666666666), ((sin(x) + (pow(sin(x), 3.0) / t_0)) / cos(x))), t_2), 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(eps, fma(eps, fma(eps, Float64(Float64(Float64(Float64(t_1 + Float64((sin(x) ^ 4.0) / t_0)) / t_0) - fma(0.16666666666666666, t_2, fma(-0.5, t_2, -0.5))) + -0.16666666666666666), Float64(Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_0)) / cos(x))), t_2), 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[(eps * N[(eps * N[(eps * N[(N[(N[(N[(t$95$1 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(0.16666666666666666 * t$95$2 + N[(-0.5 * t$95$2 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + eps), $MachinePrecision]]]]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return fma(eps, fma(Float64(sin(x) / fma(0.5, cos(Float64(x + x)), 0.5)), sin(x), Float64(eps * fma(eps, 0.3333333333333333, Float64(Float64(sin(x) * Float64((tan(x) ^ 2.0) + 1.0)) / cos(x))))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Sin[x], $MachinePrecision] / N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(eps * N[(eps * 0.3333333333333333 + N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.4%

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

12. Applied rewrites 99.3%

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

13. Final simplification 99.3%

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

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \frac{\mathsf{fma}\left(x, 0.8333333333333334 \cdot \left(x \cdot x\right), x\right)}{\cos x}\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   (sin x)
   (/ (sin x) (+ 0.5 (* 0.5 (cos (+ x x)))))
   (*
    eps
    (fma
     eps
     0.3333333333333333
     (/ (fma x (* 0.8333333333333334 (* x x)) x) (cos x)))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(sin(x), (sin(x) / (0.5 + (0.5 * cos((x + x))))), (eps * fma(eps, 0.3333333333333333, (fma(x, (0.8333333333333334 * (x * x)), x) / cos(x))))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(sin(x), Float64(sin(x) / Float64(0.5 + Float64(0.5 * cos(Float64(x + x))))), Float64(eps * fma(eps, 0.3333333333333333, Float64(fma(x, Float64(0.8333333333333334 * Float64(x * x)), x) / cos(x))))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * 0.3333333333333333 + N[(N[(x * N[(0.8333333333333334 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{3}, \frac{x \cdot \left(1 + \frac{5}{6} \cdot {x}^{2}\right)}{\cos x}\right)\right), \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333, \frac{\mathsf{fma}\left(x, 0.8333333333333334 \cdot \left(x \cdot x\right), x\right)}{\cos x}\right)\right), \varepsilon\right) \]
14. Add Preprocessing

Alternative 5: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 1.3333333333333333, x + \varepsilon, 1\right), \varepsilon \cdot 0.3333333333333333\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   (sin x)
   (/ (sin x) (+ 0.5 (* 0.5 (cos (+ x x)))))
   (*
    eps
    (fma
     x
     (fma (* x 1.3333333333333333) (+ x eps) 1.0)
     (* eps 0.3333333333333333))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(sin(x), (sin(x) / (0.5 + (0.5 * cos((x + x))))), (eps * fma(x, fma((x * 1.3333333333333333), (x + eps), 1.0), (eps * 0.3333333333333333)))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(sin(x), Float64(sin(x) / Float64(0.5 + Float64(0.5 * cos(Float64(x + x))))), Float64(eps * fma(x, fma(Float64(x * 1.3333333333333333), Float64(x + eps), 1.0), Float64(eps * 0.3333333333333333)))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * N[(N[(x * 1.3333333333333333), $MachinePrecision] * N[(x + eps), $MachinePrecision] + 1.0), $MachinePrecision] + N[(eps * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.9%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 1.3333333333333333, \varepsilon + x, 1\right), 0.3333333333333333 \cdot \varepsilon\right)\right), \varepsilon\right) \]

14. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\sin x, \frac{\sin x}{0.5 + 0.5 \cdot \cos \left(x + x\right)}, \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot 1.3333333333333333, x + \varepsilon, 1\right), \varepsilon \cdot 0.3333333333333333\right)\right), \varepsilon\right) \]
15. Add Preprocessing

Alternative 6: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333}, x\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.8%

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

Alternative 7: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{0.3333333333333333}, x\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 98.8%

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

Alternative 8: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 9: 98.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.3%

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

Alternative 10: 98.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

10. Applied rewrites 97.9%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.3%

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

14. Final simplification 98.3%

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

Alternative 11: 98.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.3%

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

9. Final simplification 98.3%

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

Alternative 12: 98.3% 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 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

Alternative 13: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

10. Applied rewrites 97.9%

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

11. Taylor expanded in x around 0

    \[\leadsto \left(1 + {x}^{2}\right) \cdot \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 97.9%

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

14. Final simplification 97.9%

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

Alternative 14: 97.9% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * 1.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * 1.0;
end

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

10. Applied rewrites 97.9%

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

11. Taylor expanded in x around 0

    \[\leadsto 1 \cdot \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 97.2%

    \[\leadsto 1 \cdot \varepsilon \]

14. Final simplification 97.2%

    \[\leadsto \varepsilon \cdot 1 \]
15. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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

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

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