2tan (problem 3.3.2)

Percentage Accurate: 61.9% → 99.6%

Time: 13.2s

Alternatives: 14

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (fma (/ (sin x) t_0) (sin x) 1.0))
        (t_2
         (fma
          t_1
          -0.3333333333333333
          (/ (fma (sin x) (sin x) (/ (pow (sin x) 4.0) t_0)) (- t_0)))))
   (*
    (fma
     (*
      (fma
       (* (sin x) (fma (/ t_1 (cos x)) -0.3333333333333333 (/ t_2 (cos x))))
       eps
       t_2)
      (- eps))
     eps
     (* t_1 (+ (/ (* (sin x) eps) (cos x)) 1.0)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = fma((sin(x) / t_0), sin(x), 1.0);
	double t_2 = fma(t_1, -0.3333333333333333, (fma(sin(x), sin(x), (pow(sin(x), 4.0) / t_0)) / -t_0));
	return fma((fma((sin(x) * fma((t_1 / cos(x)), -0.3333333333333333, (t_2 / cos(x)))), eps, t_2) * -eps), eps, (t_1 * (((sin(x) * eps) / cos(x)) + 1.0))) * eps;
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = fma(Float64(sin(x) / t_0), sin(x), 1.0)
	t_2 = fma(t_1, -0.3333333333333333, Float64(fma(sin(x), sin(x), Float64((sin(x) ^ 4.0) / t_0)) / Float64(-t_0)))
	return Float64(fma(Float64(fma(Float64(sin(x) * fma(Float64(t_1 / cos(x)), -0.3333333333333333, Float64(t_2 / cos(x)))), eps, t_2) * Float64(-eps)), eps, Float64(t_1 * Float64(Float64(Float64(sin(x) * eps) / cos(x)) + 1.0))) * 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[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * -0.3333333333333333 + 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]}, N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$1 / N[Cos[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(t$95$2 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * (-eps)), $MachinePrecision] * eps + N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (fma (/ (sin x) t_0) (sin x) 1.0)))
   (*
    (fma
     (*
      (-
       (/ (fma (sin x) (sin x) (/ (pow (sin x) 4.0) t_0)) t_0)
       (* t_1 -0.3333333333333333))
      eps)
     eps
     (* t_1 (+ (/ (* (sin x) eps) (cos x)) 1.0)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = fma((sin(x) / t_0), sin(x), 1.0);
	return fma((((fma(sin(x), sin(x), (pow(sin(x), 4.0) / t_0)) / t_0) - (t_1 * -0.3333333333333333)) * eps), eps, (t_1 * (((sin(x) * eps) / cos(x)) + 1.0))) * eps;
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = fma(Float64(sin(x) / t_0), sin(x), 1.0)
	return Float64(fma(Float64(Float64(Float64(fma(sin(x), sin(x), Float64((sin(x) ^ 4.0) / t_0)) / t_0) - Float64(t_1 * -0.3333333333333333)) * eps), eps, Float64(t_1 * Float64(Float64(Float64(sin(x) * eps) / cos(x)) + 1.0))) * 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[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[(N[(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] - N[(t$95$1 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(t$95$1 * N[(N[(N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Initial program 61.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.9%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot {\left(\cos \left(\varepsilon + x\right) \cdot \cos x\right)}^{-1} \end{array} \]

FPCore

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

C

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

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)) ** (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \sin \varepsilon \cdot {\left(\cos \left(\varepsilon + x\right) \cdot \cos x\right)}^{-1} \]
9. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

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

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(Float64(sin(eps) / 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[(N[Sin[eps], $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 0.6× speedup

Math

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

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(Float64(sin(eps) / 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[(N[Sin[eps], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

Alternative 6: 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 61.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 61.3

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 61.3

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

4. Applied rewrites 61.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-neg.f64 N/A

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

   6. distribute-frac-neg2 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. sub-neg N/A

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

   11. lift-tan.f64 N/A

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

   12. tan-quot N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. tan-quot N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. frac-sub N/A

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

6. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-lft-identity 99.9

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

8. Applied rewrites 99.9%

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

Alternative 7: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.6%

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

Alternative 8: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\cos \left(-2 \cdot x\right) + 1} \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 61.3

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 61.3

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

4. Applied rewrites 61.3%

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. metadata-eval N/A

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

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

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

   6. cos-neg N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 98.9

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

9. Applied rewrites 98.9%

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

Alternative 9: 98.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

Alternative 10: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

Alternative 11: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.6%

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

Alternative 12: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.6%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 97.6%

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

Alternative 13: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. sin-diff N/A

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

   10. lower-sin.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 N/A

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

   21. lower-cos.f64 61.4

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

4. Applied rewrites 61.4%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.6%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 6.7%

    \[\leadsto \left(x \cdot x\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 14: 5.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 61.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-cos.f64 61.3

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 61.3

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

4. Applied rewrites 61.3%

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

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 5.4

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

7. Applied rewrites 5.4%

   \[\leadsto \color{blue}{0} \]
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 2024321 
(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)))