tan-example (used to crash)

Percentage Accurate: 78.8% → 99.7%

Time: 33.3s

Alternatives: 16

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, -\tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma (/ 1.0 (- 1.0 (* (tan y) (tan z)))) (+ (tan y) (tan z)) (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma((1.0 / (1.0 - (tan(y) * tan(z)))), (tan(y) + tan(z)), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), Float64(tan(y) + tan(z)), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, -\tan a\right)} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= (tan a) -0.005)
     t_1
     (if (<= (tan a) 5e-7)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan y) (tan z))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = t_1;
	} else if (tan(a) <= 5e-7) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = t_1;
	elseif (tan(a) <= 5e-7)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-7], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 4.99999999999999977e-7 < (tan.f64 a)

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. tan-sum N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-tan.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-tan.f64 N/A

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

      18. lower-tan.f64 N/A

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

      19. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.9%

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, -\tan a\right) \]

   if -0.0050000000000000001 < (tan.f64 a) < 4.99999999999999977e-7

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.3

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. tan-sum N/A

         \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y} + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      3. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \color{blue}{\tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      5. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y} \cdot \tan z} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      6. lift-tan.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \color{blue}{\tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lift--.f64 N/A

         \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lift-/.f64 99.6

         \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) \]

   7. Applied rewrites 99.6%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (fma (tan z) (- (tan y)) 1.0)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / fma(tan(z), -tan(y), 1.0)) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / fma(tan(z), Float64(-tan(y)), 1.0)) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[z], $MachinePrecision] * (-N[Tan[y], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. tan-sum N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-tan.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-tan.f64 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\mathsf{fma}\left(\tan z, \mathsf{neg}\left(\tan y\right), 1\right)}} - \tan a\right) \]

   10. lower-neg.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)}} - \tan a\right) \]
7. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. tan-sum N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-tan.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-tan.f64 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \mathsf{fma}\left(1, t\_0, -\tan a\right)\\ \mathbf{if}\;a \leq -0.0098:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.0066:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, t\_0, x - \mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (fma 1.0 t_0 (- (tan a))))))
   (if (<= a -0.0098)
     t_1
     (if (<= a 0.0066)
       (fma
        (/ 1.0 (- 1.0 (* (tan y) (tan z))))
        t_0
        (- x (fma a (* (* a a) 0.3333333333333333) a)))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + fma(1.0, t_0, -tan(a));
	double tmp;
	if (a <= -0.0098) {
		tmp = t_1;
	} else if (a <= 0.0066) {
		tmp = fma((1.0 / (1.0 - (tan(y) * tan(z)))), t_0, (x - fma(a, ((a * a) * 0.3333333333333333), a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + fma(1.0, t_0, Float64(-tan(a))))
	tmp = 0.0
	if (a <= -0.0098)
		tmp = t_1;
	elseif (a <= 0.0066)
		tmp = fma(Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), t_0, Float64(x - fma(a, Float64(Float64(a * a) * 0.3333333333333333), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(1.0 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0098], t$95$1, If[LessEqual[a, 0.0066], N[(N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(x - N[(a * N[(N[(a * a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0097999999999999997 or 0.0066 < a

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. tan-sum N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-tan.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-tan.f64 N/A

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

      18. lower-tan.f64 N/A

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

      19. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.9%

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, -\tan a\right) \]

   if -0.0097999999999999997 < a < 0.0066

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.3

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, \left(-\mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right) + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0098:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \mathbf{elif}\;a \leq 0.0066:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{1 - \tan y \cdot \tan z}, \tan y + \tan z, x - \mathsf{fma}\left(a, \left(a \cdot a\right) \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.005)
   (+ x (- (tan y) (tan a)))
   (if (<= (tan a) 2e-14)
     (fma
      (/ (fma a (fma (* a a) -0.3333333333333333 -1.0) (tan (+ y z))) x)
      x
      x)
     (+ (tan y) (- x (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x + (tan(y) - tan(a));
	} else if (tan(a) <= 2e-14) {
		tmp = fma((fma(a, fma((a * a), -0.3333333333333333, -1.0), tan((y + z))) / x), x, x);
	} else {
		tmp = tan(y) + (x - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	elseif (tan(a) <= 2e-14)
		tmp = fma(Float64(fma(a, fma(Float64(a * a), -0.3333333333333333, -1.0), tan(Float64(y + z))) / x), x, x);
	else
		tmp = Float64(tan(y) + Float64(x - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-14], N[(N[(N[(a * N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[Tan[y], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 74.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.2

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

   5. Applied rewrites 54.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 54.2

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 54.2

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

   7. Applied rewrites 54.2%

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

   if -0.0050000000000000001 < (tan.f64 a) < 2e-14

   1. Initial program 82.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 82.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)}\right) - \left(\frac{1}{3} \cdot \frac{{a}^{3}}{x} + \frac{a}{x}\right)\right)} \]

   8. Applied rewrites 82.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \left(\frac{-1}{3} \cdot \color{blue}{\left(a \cdot a\right)} + -1\right) + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)} + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\color{blue}{\sin \left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\cos \color{blue}{\left(z + y\right)}}}{x}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\color{blue}{\cos \left(z + y\right)}}}{x}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}}{x}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}}{x}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}}\right) \]

   10. Applied rewrites 82.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)} \]

   if 2e-14 < (tan.f64 a)

   1. Initial program 78.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.9

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

   5. Applied rewrites 54.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 N/A

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

      13. associate-+l- N/A

          \[\leadsto \color{blue}{\tan y - \left(\tan a - x\right)} \]

      14. lower--.f64 N/A

          \[\leadsto \color{blue}{\tan y - \left(\tan a - x\right)} \]

      15. lower--.f64 54.9

          \[\leadsto \tan y - \color{blue}{\left(\tan a - x\right)} \]

   7. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\tan y - \left(\tan a - x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -0.005)
     t_0
     (if (<= (tan a) 2e-14)
       (fma
        (/ (fma a (fma (* a a) -0.3333333333333333 -1.0) (tan (+ y z))) x)
        x
        x)
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(y) - tan(a));
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = t_0;
	} else if (tan(a) <= 2e-14) {
		tmp = fma((fma(a, fma((a * a), -0.3333333333333333, -1.0), tan((y + z))) / x), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = t_0;
	elseif (tan(a) <= 2e-14)
		tmp = fma(Float64(fma(a, fma(Float64(a * a), -0.3333333333333333, -1.0), tan(Float64(y + z))) / x), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.005], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-14], N[(N[(N[(a * N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0050000000000000001 or 2e-14 < (tan.f64 a)

   1. Initial program 76.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.5

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

   5. Applied rewrites 54.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 54.5

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 54.6

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

   7. Applied rewrites 54.6%

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

   if -0.0050000000000000001 < (tan.f64 a) < 2e-14

   1. Initial program 82.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 82.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 82.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)}\right) - \left(\frac{1}{3} \cdot \frac{{a}^{3}}{x} + \frac{a}{x}\right)\right)} \]

   8. Applied rewrites 82.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \left(\frac{-1}{3} \cdot \color{blue}{\left(a \cdot a\right)} + -1\right) + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)} + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\color{blue}{\sin \left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\cos \color{blue}{\left(z + y\right)}}}{x}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\color{blue}{\cos \left(z + y\right)}}}{x}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}}{x}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}}{x}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}}\right) \]

   10. Applied rewrites 82.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(1, \tan y + \tan z, -\tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (fma 1.0 (+ (tan y) (tan z)) (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma(1.0, (tan(y) + tan(z)), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(1.0, Float64(tan(y) + tan(z)), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(1.0 * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. tan-sum N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-+.f64 N/A

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

   17. lower-tan.f64 N/A

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

   18. lower-tan.f64 N/A

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

   19. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

   \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, \mathsf{neg}\left(\tan a\right)\right) \]
6. Step-by-step derivation

7. Applied rewrites 79.5%

   \[\leadsto x + \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, -\tan a\right) \]
8. Add Preprocessing

Alternative 9: 78.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 10: 59.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.4)
     t_0
     (if (<= a 0.22)
       (fma
        (/ (fma a (fma (* a a) -0.3333333333333333 -1.0) (tan (+ y z))) x)
        x
        x)
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.4) {
		tmp = t_0;
	} else if (a <= 0.22) {
		tmp = fma((fma(a, fma((a * a), -0.3333333333333333, -1.0), tan((y + z))) / x), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.4)
		tmp = t_0;
	elseif (a <= 0.22)
		tmp = fma(Float64(fma(a, fma(Float64(a * a), -0.3333333333333333, -1.0), tan(Float64(y + z))) / x), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.4], t$95$0, If[LessEqual[a, 0.22], N[(N[(N[(a * N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.40000000000000002 or 0.220000000000000001 < a

   1. Initial program 77.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

      2. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

      3. lower-sin.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

      4. lower-cos.f64 41.0

         \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

   8. Applied rewrites 41.0%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      2. lift-tan.f64 N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      3. lower--.f64 41.0

         \[\leadsto \color{blue}{x - \tan a} \]

   10. Applied rewrites 41.0%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -0.40000000000000002 < a < 0.220000000000000001

   1. Initial program 81.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)}\right) - \left(\frac{1}{3} \cdot \frac{{a}^{3}}{x} + \frac{a}{x}\right)\right)} \]

   8. Applied rewrites 81.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \left(\frac{-1}{3} \cdot \color{blue}{\left(a \cdot a\right)} + -1\right) + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)} + \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}{x}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\color{blue}{\sin \left(z + y\right)}}{\cos \left(z + y\right)}}{x}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\cos \color{blue}{\left(z + y\right)}}}{x}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \frac{\sin \left(z + y\right)}{\color{blue}{\cos \left(z + y\right)}}}{x}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right) + \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}}}{x}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \frac{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}}{x}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}}\right) \]

   10. Applied rewrites 81.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), \tan \left(y + z\right)\right)}{x}, x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.36:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.7)
     t_0
     (if (<= a 0.36)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 0.13333333333333333) 0.3333333333333333)
          (* a (* a a))
          a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.7) {
		tmp = t_0;
	} else if (a <= 0.36) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.7)
		tmp = t_0;
	elseif (a <= 0.36)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.7], t$95$0, If[LessEqual[a, 0.36], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.69999999999999996 or 0.35999999999999999 < a

   1. Initial program 77.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

      2. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

      3. lower-sin.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

      4. lower-cos.f64 41.0

         \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

   8. Applied rewrites 41.0%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      2. lift-tan.f64 N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      3. lower--.f64 41.0

         \[\leadsto \color{blue}{x - \tan a} \]

   10. Applied rewrites 41.0%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -0.69999999999999996 < a < 0.35999999999999999

   1. Initial program 81.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot {a}^{2}\right) \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}, a \cdot {a}^{2}, a\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {a}^{2} + \frac{1}{3}}, a \cdot {a}^{2}, a\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \frac{2}{15}} + \frac{1}{3}, a \cdot {a}^{2}, a\right)\right) \]

      9. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{2}{15} + \frac{1}{3}, a \cdot {a}^{2}, a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot \left(a \cdot \frac{2}{15}\right)} + \frac{1}{3}, a \cdot {a}^{2}, a\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a, a \cdot \frac{2}{15}, \frac{1}{3}\right)}, a \cdot {a}^{2}, a\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, \color{blue}{a \cdot \frac{2}{15}}, \frac{1}{3}\right), a \cdot {a}^{2}, a\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \frac{2}{15}, \frac{1}{3}\right), \color{blue}{a \cdot {a}^{2}}, a\right)\right) \]

      14. unpow2 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \frac{2}{15}, \frac{1}{3}\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      15. lower-*.f64 81.2

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   5. Applied rewrites 81.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.22:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x + \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.4)
     t_0
     (if (<= a 0.22)
       (fma (fma (* a a) -0.3333333333333333 -1.0) a (+ x (tan (+ y z))))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.4) {
		tmp = t_0;
	} else if (a <= 0.22) {
		tmp = fma(fma((a * a), -0.3333333333333333, -1.0), a, (x + tan((y + z))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.4)
		tmp = t_0;
	elseif (a <= 0.22)
		tmp = fma(fma(Float64(a * a), -0.3333333333333333, -1.0), a, Float64(x + tan(Float64(y + z))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.4], t$95$0, If[LessEqual[a, 0.22], N[(N[(N[(a * a), $MachinePrecision] * -0.3333333333333333 + -1.0), $MachinePrecision] * a + N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.40000000000000002 or 0.220000000000000001 < a

   1. Initial program 77.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 54.6

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

      2. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

      3. lower-sin.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

      4. lower-cos.f64 41.0

         \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

   8. Applied rewrites 41.0%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      2. lift-tan.f64 N/A

         \[\leadsto x - \color{blue}{\tan a} \]

      3. lower--.f64 41.0

         \[\leadsto \color{blue}{x - \tan a} \]

   10. Applied rewrites 41.0%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -0.40000000000000002 < a < 0.220000000000000001

   1. Initial program 81.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

      9. lower-*.f64 81.2

         \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

   5. Applied rewrites 81.2%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{3} \cdot \left(\left(\frac{x}{{a}^{3}} + \frac{\sin \left(y + z\right)}{{a}^{3} \cdot \cos \left(y + z\right)}\right) - \left(\frac{1}{3} + \frac{1}{{a}^{2}}\right)\right)} \]

   8. Applied rewrites 24.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right)\right)} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \color{blue}{\left(z + y\right)}}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      2. lift-sin.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\color{blue}{\sin \left(z + y\right)}}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \color{blue}{\left(z + y\right)}} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\cos \left(z + y\right)}} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\color{blue}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)}} + \frac{x}{a \cdot \left(a \cdot a\right)}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)} + \frac{x}{a \cdot \color{blue}{\left(a \cdot a\right)}}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \frac{\sin \left(z + y\right)}{\left(a \cdot \left(a \cdot a\right)\right) \cdot \cos \left(z + y\right)} + \frac{x}{\color{blue}{a \cdot \left(a \cdot a\right)}}, a \cdot \mathsf{fma}\left(\frac{-1}{3}, a \cdot a, -1\right)\right) \]

   10. Applied rewrites 24.0%

       \[\leadsto \mathsf{fma}\left(a \cdot \left(a \cdot a\right), \color{blue}{\mathsf{fma}\left(\frac{1}{a \cdot \left(a \cdot a\right)}, \tan \left(y + z\right), \frac{x}{a \cdot \left(a \cdot a\right)}\right)}, a \cdot \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right)\right) \]

   12. Applied rewrites 81.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.3333333333333333, -1\right), a, x + \tan \left(y + z\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \tan a \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (- x (tan a)))

C

double code(double x, double y, double z, double a) {
	return x - tan(a);
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x - tan(a)
end function

Java

public static double code(double x, double y, double z, double a) {
	return x - Math.tan(a);
}

Python

def code(x, y, z, a):
	return x - math.tan(a)

Julia

function code(x, y, z, a)
	return Float64(x - tan(a))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - tan(a);
end

Wolfram

code[x_, y_, z_, a_] := N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

   2. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

   3. lower-sin.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

   4. lower-cos.f64 38.5

      \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

8. Applied rewrites 38.5%

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
9. Step-by-step derivation

   1. tan-quot N/A

      \[\leadsto x - \color{blue}{\tan a} \]

   2. lift-tan.f64 N/A

      \[\leadsto x - \color{blue}{\tan a} \]

   3. lower--.f64 38.5

      \[\leadsto \color{blue}{x - \tan a} \]

10. Applied rewrites 38.5%

    \[\leadsto \color{blue}{x - \tan a} \]
11. Add Preprocessing

Alternative 14: 22.2% accurate, 52.5× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (- x a))

C

double code(double x, double y, double z, double a) {
	return x - a;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x - a
end function

Java

public static double code(double x, double y, double z, double a) {
	return x - a;
}

Python

def code(x, y, z, a):
	return x - a

Julia

function code(x, y, z, a)
	return Float64(x - a)
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - a;
end

Wolfram

code[x_, y_, z_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

   2. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

   3. lower-sin.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

   4. lower-cos.f64 38.5

      \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

8. Applied rewrites 38.5%

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + -1 \cdot a} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

       \[\leadsto \color{blue}{x - a} \]

    3. lower--.f64 19.6

       \[\leadsto \color{blue}{x - a} \]

11. Applied rewrites 19.6%

    \[\leadsto \color{blue}{x - a} \]
12. Add Preprocessing

Alternative 15: 3.5% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (- a))

C

double code(double x, double y, double z, double a) {
	return -a;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = -a
end function

Java

public static double code(double x, double y, double z, double a) {
	return -a;
}

Python

def code(x, y, z, a):
	return -a

Julia

function code(x, y, z, a)
	return Float64(-a)
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = -a;
end

Wolfram

code[x_, y_, z_, a_] := (-a)

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto x + \left(\color{blue}{\frac{\sin y}{\cos y}} - \tan a\right) \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 55.7

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

5. Applied rewrites 55.7%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

   2. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{\sin a}{\cos a}} \]

   3. lower-sin.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\sin a}}{\cos a} \]

   4. lower-cos.f64 38.5

      \[\leadsto x - \frac{\sin a}{\color{blue}{\cos a}} \]

8. Applied rewrites 38.5%

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{x + -1 \cdot a} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

       \[\leadsto \color{blue}{x - a} \]

    3. lower--.f64 19.6

       \[\leadsto \color{blue}{x - a} \]

11. Applied rewrites 19.6%

    \[\leadsto \color{blue}{x - a} \]

12. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-1 \cdot a} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

    2. lower-neg.f64 3.7

       \[\leadsto \color{blue}{-a} \]

14. Applied rewrites 3.7%

    \[\leadsto \color{blue}{-a} \]
15. Add Preprocessing

Alternative 16: 31.6% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 x)

C

double code(double x, double y, double z, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 79.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \frac{1}{3} \cdot {a}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\color{blue}{a \cdot a}, a \cdot \frac{1}{3}, a\right)\right) \]

   9. lower-*.f64 41.6

      \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot 0.3333333333333333}, a\right)\right) \]

5. Applied rewrites 41.6%

   \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)}\right) - \left(\frac{1}{3} \cdot \frac{{a}^{3}}{x} + \frac{a}{x}\right)\right)} \]

8. Applied rewrites 41.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.3333333333333333, a \cdot a, -1\right), \frac{\sin \left(z + y\right)}{\cos \left(z + y\right)}\right)}{x}\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 29.0%

    \[\leadsto x \cdot \color{blue}{1} \]

12. Final simplification 29.0%

    \[\leadsto x \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))