tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 20.4s

Alternatives: 11

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: 79.3% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan z \cdot \tan y\\ x + \left(\frac{\tan y}{t\_0} + \left(\frac{\tan z}{t\_0} - \tan a\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan z) (tan y)))))
   (+ x (+ (/ (tan y) t_0) (- (/ (tan z) t_0) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(z) * tan(y));
	return x + ((tan(y) / t_0) + ((tan(z) / t_0) - 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
    real(8) :: t_0
    t_0 = 1.0d0 - (tan(z) * tan(y))
    code = x + ((tan(y) / t_0) + ((tan(z) / t_0) - tan(a)))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (Math.tan(z) * Math.tan(y));
	return x + ((Math.tan(y) / t_0) + ((Math.tan(z) / t_0) - Math.tan(a)));
}

Python

def code(x, y, z, a):
	t_0 = 1.0 - (math.tan(z) * math.tan(y))
	return x + ((math.tan(y) / t_0) + ((math.tan(z) / t_0) - math.tan(a)))

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 - Float64(tan(z) * tan(y)))
	return Float64(x + Float64(Float64(tan(y) / t_0) + Float64(Float64(tan(z) / t_0) - tan(a))))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(z) * tan(y));
	tmp = x + ((tan(y) / t_0) + ((tan(z) / t_0) - tan(a)));
end

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. tan-sum N/A

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

   5. div-add N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-tan.f64 N/A

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

   15. lower--.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-34}:\\ \;\;\;\;-\mathsf{fma}\left(-\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot x} + 1\right), x, a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.002)
   (fma (/ (- (/ (sin (+ z y)) (cos (+ z y))) (/ (sin a) (cos a))) x) x x)
   (if (<= (tan a) 4e-34)
     (-
      (fma
       (- (+ (/ (+ (tan y) (tan z)) (* (fma (- (tan z)) (tan y) 1.0) x)) 1.0))
       x
       a))
     (+ x (- (tan (+ y z)) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.002) {
		tmp = fma((((sin((z + y)) / cos((z + y))) - (sin(a) / cos(a))) / x), x, x);
	} else if (tan(a) <= 4e-34) {
		tmp = -fma(-(((tan(y) + tan(z)) / (fma(-tan(z), tan(y), 1.0) * x)) + 1.0), x, a);
	} else {
		tmp = x + (tan((y + z)) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.002)
		tmp = fma(Float64(Float64(Float64(sin(Float64(z + y)) / cos(Float64(z + y))) - Float64(sin(a) / cos(a))) / x), x, x);
	elseif (tan(a) <= 4e-34)
		tmp = Float64(-fma(Float64(-Float64(Float64(Float64(tan(y) + tan(z)) / Float64(fma(Float64(-tan(z)), tan(y), 1.0) * x)) + 1.0)), x, a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.002], N[(N[(N[(N[(N[Sin[N[(z + y), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 4e-34], (-N[((-N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]) * x + a), $MachinePrecision]), N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -2e-3

   1. Initial program 90.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 90.6%

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

   if -2e-3 < (tan.f64 a) < 3.99999999999999971e-34

   1. Initial program 81.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.5%

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

   10. Applied rewrites 99.5%

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

   if 3.99999999999999971e-34 < (tan.f64 a)

   1. Initial program 81.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-34}:\\ \;\;\;\;-\mathsf{fma}\left(-\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot x} + 1\right), x, a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - 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(z) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.00018 \lor \neg \left(a \leq 10^{-31}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(-\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot x} + 1\right), x, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -0.00018) (not (<= a 1e-31)))
   (+ x (- (tan (+ y z)) (tan a)))
   (-
    (fma
     (- (+ (/ (+ (tan y) (tan z)) (* (fma (- (tan z)) (tan y) 1.0) x)) 1.0))
     x
     a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.00018) || !(a <= 1e-31)) {
		tmp = x + (tan((y + z)) - tan(a));
	} else {
		tmp = -fma(-(((tan(y) + tan(z)) / (fma(-tan(z), tan(y), 1.0) * x)) + 1.0), x, a);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.00018) || !(a <= 1e-31))
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	else
		tmp = Float64(-fma(Float64(-Float64(Float64(Float64(tan(y) + tan(z)) / Float64(fma(Float64(-tan(z)), tan(y), 1.0) * x)) + 1.0)), x, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -0.00018], N[Not[LessEqual[a, 1e-31]], $MachinePrecision]], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[((-N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]) * x + a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if a < -1.80000000000000011e-4 or 1e-31 < a

   1. Initial program 86.0%

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

   if -1.80000000000000011e-4 < a < 1e-31

   1. Initial program 81.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.5%

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

   10. Applied rewrites 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00018 \lor \neg \left(a \leq 10^{-31}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(-\left(\frac{\tan y + \tan z}{\mathsf{fma}\left(-\tan z, \tan y, 1\right) \cdot x} + 1\right), x, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% 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 84.0%

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

Alternative 6: 50.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.55) (not (<= a 1.55)))
   (- (- x))
   (-
    (tan (+ z y))
    (-
     (*
      (fma
       (fma
        (fma (* a a) 0.05396825396825397 0.13333333333333333)
        (* a a)
        0.3333333333333333)
       (* a a)
       1.0)
      a)
     x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.55) || !(a <= 1.55)) {
		tmp = -(-x);
	} else {
		tmp = tan((z + y)) - ((fma(fma(fma((a * a), 0.05396825396825397, 0.13333333333333333), (a * a), 0.3333333333333333), (a * a), 1.0) * a) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -1.55) || !(a <= 1.55))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(fma(fma(Float64(a * a), 0.05396825396825397, 0.13333333333333333), Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.55], N[Not[LessEqual[a, 1.55]], $MachinePrecision]], (-(-x)), N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 1.55000000000000004 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 3.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto --1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 22.8%

       \[\leadsto -\left(-x\right) \]

   if -1.55000000000000004 < a < 1.55000000000000004

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)} - x\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a} - x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a} - x\right) \]

      3. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) + 1\right)} \cdot a - x\right) \]

      4. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\left(\color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot {a}^{2}} + 1\right) \cdot a - x\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right), {a}^{2}, 1\right)} \cdot a - x\right) \]

      6. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right) + \frac{1}{3}}, {a}^{2}, 1\right) \cdot a - x\right) \]

      7. *-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right) \cdot {a}^{2}} + \frac{1}{3}, {a}^{2}, 1\right) \cdot a - x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}, {a}^{2}, \frac{1}{3}\right)}, {a}^{2}, 1\right) \cdot a - x\right) \]

      9. +-commutative N/A

         \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{17}{315} \cdot {a}^{2} + \frac{2}{15}}, {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      10. *-commutative N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{a}^{2} \cdot \frac{17}{315}} + \frac{2}{15}, {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({a}^{2}, \frac{17}{315}, \frac{2}{15}\right)}, {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      12. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{17}{315}, \frac{2}{15}\right), {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{17}{315}, \frac{2}{15}\right), {a}^{2}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      14. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \frac{17}{315}, \frac{2}{15}\right), \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \frac{17}{315}, \frac{2}{15}\right), \color{blue}{a \cdot a}, \frac{1}{3}\right), {a}^{2}, 1\right) \cdot a - x\right) \]

      16. unpow2 N/A

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, \frac{17}{315}, \frac{2}{15}\right), a \cdot a, \frac{1}{3}\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

      17. lower-*.f64 83.1

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

   7. Applied rewrites 83.1%

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a} - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.55) (not (<= a 1.55)))
   (- (- x))
   (-
    (tan (+ z y))
    (-
     (*
      (fma (fma (* a a) 0.13333333333333333 0.3333333333333333) (* a a) 1.0)
      a)
     x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.55) || !(a <= 1.55)) {
		tmp = -(-x);
	} else {
		tmp = tan((z + y)) - ((fma(fma((a * a), 0.13333333333333333, 0.3333333333333333), (a * a), 1.0) * a) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -1.55) || !(a <= 1.55))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(fma(Float64(a * a), 0.13333333333333333, 0.3333333333333333), Float64(a * a), 1.0) * a) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.55], N[Not[LessEqual[a, 1.55]], $MachinePrecision]], (-(-x)), N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 1.55000000000000004 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 3.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto --1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 22.8%

       \[\leadsto -\left(-x\right) \]

   if -1.55000000000000004 < a < 1.55000000000000004

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.9

          \[\leadsto \tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), \color{blue}{a \cdot a}, 1\right) \cdot a - x\right) \]

   7. Applied rewrites 82.9%

      \[\leadsto \tan \left(z + y\right) - \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a} - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.55) (not (<= a 1.55)))
   (- (- x))
   (- (tan (+ z y)) (- (* (fma (* a a) 0.3333333333333333 1.0) a) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.55) || !(a <= 1.55)) {
		tmp = -(-x);
	} else {
		tmp = tan((z + y)) - ((fma((a * a), 0.3333333333333333, 1.0) * a) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -1.55) || !(a <= 1.55))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.55], N[Not[LessEqual[a, 1.55]], $MachinePrecision]], (-(-x)), N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 1.55000000000000004 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 3.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto --1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 22.8%

       \[\leadsto -\left(-x\right) \]

   if -1.55000000000000004 < a < 1.55000000000000004

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 82.7

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

   7. Applied rewrites 82.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= a -1.55) (not (<= a 1.55))) (- (- x)) (- (tan (+ z y)) (- a x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.55) || !(a <= 1.55)) {
		tmp = -(-x);
	} else {
		tmp = tan((z + y)) - (a - x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((a <= (-1.55d0)) .or. (.not. (a <= 1.55d0))) then
        tmp = -(-x)
    else
        tmp = tan((z + y)) - (a - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -1.55) || !(a <= 1.55)) {
		tmp = -(-x);
	} else {
		tmp = Math.tan((z + y)) - (a - x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (a <= -1.55) or not (a <= 1.55):
		tmp = -(-x)
	else:
		tmp = math.tan((z + y)) - (a - x)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -1.55) || !(a <= 1.55))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(a - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((a <= -1.55) || ~((a <= 1.55)))
		tmp = -(-x);
	else
		tmp = tan((z + y)) - (a - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[a, -1.55], N[Not[LessEqual[a, 1.55]], $MachinePrecision]], (-(-x)), N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(a - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 1.55000000000000004 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 3.5%

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

   12. Taylor expanded in x around inf

       \[\leadsto --1 \cdot x \]
   13. Step-by-step derivation

   14. Applied rewrites 22.8%

       \[\leadsto -\left(-x\right) \]

   if -1.55000000000000004 < a < 1.55000000000000004

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 82.1

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

   7. Applied rewrites 82.1%

      \[\leadsto \tan \left(z + y\right) - \color{blue}{\left(a - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.55 \lor \neg \left(a \leq 1.55\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(a - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.8% accurate, 42.0× speedup

Math

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

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 Float64(-Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 45.4%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 35.4%

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

12. Taylor expanded in x around inf

    \[\leadsto --1 \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 33.0%

    \[\leadsto -\left(-x\right) \]
15. Add Preprocessing

Alternative 11: 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 84.0%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 45.4%

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

9. Taylor expanded in a around inf

   \[\leadsto -1 \cdot a \]
10. Step-by-step derivation

11. Applied rewrites 3.7%

    \[\leadsto -a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))