tan-example (used to crash)

Percentage Accurate: 79.7% → 99.7%

Time: 27.8s

Alternatives: 15

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: 79.7% 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, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	double t_0 = fma(-tan(z), tan(y), 1.0);
	return x + ((fma(cos(a), (tan(y) + tan(z)), (-sin(a) * t_0)) / cos(a)) / t_0);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(z) * tan(y));
	tmp = x + ((((tan(z) + tan(y)) * cos(a)) - (t_0 * sin(a))) / (t_0 * cos(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[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto x + \color{blue}{\frac{\left(\tan z + \tan y\right) \cdot \cos a - \left(1 - \tan z \cdot \tan y\right) \cdot \sin a}{\left(1 - \tan z \cdot \tan y\right) \cdot \cos a}} \]
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 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0112:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.02)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= a 0.0112)
     (-
      x
      (+
       (/ (+ (tan z) (tan y)) (+ -1.0 (* (tan z) (tan y))))
       (* (fma (* a a) 0.3333333333333333 1.0) a)))
     (fma (/ (- (/ (sin (+ z y)) (cos (+ z y))) (/ (sin a) (cos a))) x) x x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.02) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 0.0112) {
		tmp = x - (((tan(z) + tan(y)) / (-1.0 + (tan(z) * tan(y)))) + (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = fma((((sin((z + y)) / cos((z + y))) - (sin(a) / cos(a))) / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 0.0112)
		tmp = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / Float64(-1.0 + Float64(tan(z) * tan(y)))) + Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = fma(Float64(Float64(Float64(sin(Float64(z + y)) / cos(Float64(z + y))) - Float64(sin(a) / cos(a))) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.02], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0112], 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[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0200000000000000004

   1. Initial program 83.1%

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

   if -0.0200000000000000004 < a < 0.0111999999999999999

   1. Initial program 75.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.0111999999999999999 < a

   1. Initial program 79.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.4%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0112:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} - \frac{\sin a}{\cos a}}{x}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0112:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\tan a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.02)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= a 0.0112)
     (-
      x
      (+
       (/ (+ (tan z) (tan y)) (+ -1.0 (* (tan z) (tan y))))
       (* (fma (* a a) 0.3333333333333333 1.0) a)))
     (- (tan (+ z y)) (- (tan a) x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.02) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 0.0112) {
		tmp = x - (((tan(z) + tan(y)) / (-1.0 + (tan(z) * tan(y)))) + (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = tan((z + y)) - (tan(a) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 0.0112)
		tmp = Float64(x - Float64(Float64(Float64(tan(z) + tan(y)) / Float64(-1.0 + Float64(tan(z) * tan(y)))) + Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(tan(a) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.02], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0112], 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[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0200000000000000004

   1. Initial program 83.1%

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

   if -0.0200000000000000004 < a < 0.0111999999999999999

   1. Initial program 75.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 0.0111999999999999999 < a

   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

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

      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 79.4

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

   4. Applied rewrites 79.4%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.0112:\\ \;\;\;\;x - \left(\frac{\tan z + \tan y}{-1 + \tan z \cdot \tan y} + \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\tan a - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)} - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z + y\right) - \left(\tan a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -2.25e-11)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= a 1.8e-11)
     (- (/ (+ (tan z) (tan y)) (fma (- (tan z)) (tan y) 1.0)) (- x))
     (- (tan (+ z y)) (- (tan a) x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -2.25e-11) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (a <= 1.8e-11) {
		tmp = ((tan(z) + tan(y)) / fma(-tan(z), tan(y), 1.0)) - -x;
	} else {
		tmp = tan((z + y)) - (tan(a) - x);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -2.25e-11)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (a <= 1.8e-11)
		tmp = Float64(Float64(Float64(tan(z) + tan(y)) / fma(Float64(-tan(z)), tan(y), 1.0)) - Float64(-x));
	else
		tmp = Float64(tan(Float64(z + y)) - Float64(tan(a) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -2.25e-11], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.8e-11], N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - (-x)), $MachinePrecision], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.25e-11

   1. Initial program 83.0%

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

   if -2.25e-11 < a < 1.79999999999999992e-11

   1. Initial program 75.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.1

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

   7. Applied rewrites 75.1%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lower-/.f64 99.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 1.79999999999999992e-11 < a

   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

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

      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 79.4

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

   4. Applied rewrites 79.4%

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

Alternative 7: 56.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(\left(\frac{z}{y} + 1\right) \cdot y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-13)
   (- (tan (* (+ (/ z y) 1.0) y)) (- x))
   (+ (- (tan z) (tan a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = tan((((z / y) + 1.0) * y)) - -x;
	} else {
		tmp = (tan(z) - tan(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 ((y + z) <= (-2d-13)) then
        tmp = tan((((z / y) + 1.0d0) * y)) - -x
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = Math.tan((((z / y) + 1.0) * y)) - -x;
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2e-13:
		tmp = math.tan((((z / y) + 1.0) * y)) - -x
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-13)
		tmp = Float64(tan(Float64(Float64(Float64(z / y) + 1.0) * y)) - Float64(-x));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2e-13)
		tmp = tan((((z / y) + 1.0) * y)) - -x;
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-13], N[(N[Tan[N[(N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -2.0000000000000001e-13

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if -2.0000000000000001e-13 < (+.f64 y z)

   1. Initial program 82.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 62.9

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

   7. Applied rewrites 62.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. lower-+.f64 N/A

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

      5. lower--.f64 62.9

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

   9. Applied rewrites 62.9%

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

Alternative 8: 79.7% 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 78.5%

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

Alternative 9: 51.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(\left(\frac{z}{y} + 1\right) \cdot y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, z \cdot z, 0.13333333333333333\right), z \cdot z, 0.3333333333333333\right), z \cdot z, 1\right) \cdot z - \left(\tan a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-13)
   (- (tan (* (+ (/ z y) 1.0) y)) (- x))
   (if (<= (+ y z) 1.0)
     (-
      (*
       (fma
        (fma
         (fma 0.05396825396825397 (* z z) 0.13333333333333333)
         (* z z)
         0.3333333333333333)
        (* z z)
        1.0)
       z)
      (- (tan a) x))
     (- (tan z) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = tan((((z / y) + 1.0) * y)) - -x;
	} else if ((y + z) <= 1.0) {
		tmp = (fma(fma(fma(0.05396825396825397, (z * z), 0.13333333333333333), (z * z), 0.3333333333333333), (z * z), 1.0) * z) - (tan(a) - x);
	} else {
		tmp = tan(z) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-13)
		tmp = Float64(tan(Float64(Float64(Float64(z / y) + 1.0) * y)) - Float64(-x));
	elseif (Float64(y + z) <= 1.0)
		tmp = Float64(Float64(fma(fma(fma(0.05396825396825397, Float64(z * z), 0.13333333333333333), Float64(z * z), 0.3333333333333333), Float64(z * z), 1.0) * z) - Float64(tan(a) - x));
	else
		tmp = Float64(tan(z) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-13], N[(N[Tan[N[(N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1.0], N[(N[(N[(N[(N[(0.05396825396825397 * N[(z * z), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(z * z), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2.0000000000000001e-13

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if -2.0000000000000001e-13 < (+.f64 y z) < 1

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.5%

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

   if 1 < (+.f64 y z)

   1. Initial program 70.3%

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

      1. lift-+.f64 N/A

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

      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 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.1

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 30.7

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

   10. Applied rewrites 30.7%

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

   12. Applied rewrites 30.7%

       \[\leadsto \color{blue}{\tan z - \left(-x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(\left(\frac{z}{y} + 1\right) \cdot y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot z, 0.13333333333333333, 0.3333333333333333\right), z \cdot z, 1\right) \cdot z - \left(\tan a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-13)
   (- (tan (* (+ (/ z y) 1.0) y)) (- x))
   (if (<= (+ y z) 1.0)
     (-
      (*
       (fma (fma (* z z) 0.13333333333333333 0.3333333333333333) (* z z) 1.0)
       z)
      (- (tan a) x))
     (- (tan z) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = tan((((z / y) + 1.0) * y)) - -x;
	} else if ((y + z) <= 1.0) {
		tmp = (fma(fma((z * z), 0.13333333333333333, 0.3333333333333333), (z * z), 1.0) * z) - (tan(a) - x);
	} else {
		tmp = tan(z) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-13)
		tmp = Float64(tan(Float64(Float64(Float64(z / y) + 1.0) * y)) - Float64(-x));
	elseif (Float64(y + z) <= 1.0)
		tmp = Float64(Float64(fma(fma(Float64(z * z), 0.13333333333333333, 0.3333333333333333), Float64(z * z), 1.0) * z) - Float64(tan(a) - x));
	else
		tmp = Float64(tan(z) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-13], N[(N[Tan[N[(N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1.0], N[(N[(N[(N[(N[(z * z), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2.0000000000000001e-13

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if -2.0000000000000001e-13 < (+.f64 y z) < 1

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.5%

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

   if 1 < (+.f64 y z)

   1. Initial program 70.3%

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

      1. lift-+.f64 N/A

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

      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 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.1

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 30.7

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

   10. Applied rewrites 30.7%

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

   12. Applied rewrites 30.7%

       \[\leadsto \color{blue}{\tan z - \left(-x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(\left(\frac{z}{y} + 1\right) \cdot y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \left(\tan a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-13)
   (- (tan (* (+ (/ z y) 1.0) y)) (- x))
   (if (<= (+ y z) 1.0)
     (- (* (fma (* z z) 0.3333333333333333 1.0) z) (- (tan a) x))
     (- (tan z) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = tan((((z / y) + 1.0) * y)) - -x;
	} else if ((y + z) <= 1.0) {
		tmp = (fma((z * z), 0.3333333333333333, 1.0) * z) - (tan(a) - x);
	} else {
		tmp = tan(z) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-13)
		tmp = Float64(tan(Float64(Float64(Float64(z / y) + 1.0) * y)) - Float64(-x));
	elseif (Float64(y + z) <= 1.0)
		tmp = Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - Float64(tan(a) - x));
	else
		tmp = Float64(tan(z) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-13], N[(N[Tan[N[(N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1.0], N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2.0000000000000001e-13

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.4

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

   7. Applied rewrites 43.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if -2.0000000000000001e-13 < (+.f64 y z) < 1

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.5%

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

   if 1 < (+.f64 y z)

   1. Initial program 70.3%

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

      1. lift-+.f64 N/A

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

      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 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.1

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 30.7

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

   10. Applied rewrites 30.7%

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

   12. Applied rewrites 30.7%

       \[\leadsto \color{blue}{\tan z - \left(-x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(z \cdot z, 0.3333333333333333, 1\right) \cdot z - \left(\tan a - x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-13)
   (- (tan (+ z y)) (- x))
   (if (<= (+ y z) 1.0)
     (- (* (fma (* z z) 0.3333333333333333 1.0) z) (- (tan a) x))
     (- (tan z) (- x)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-13) {
		tmp = tan((z + y)) - -x;
	} else if ((y + z) <= 1.0) {
		tmp = (fma((z * z), 0.3333333333333333, 1.0) * z) - (tan(a) - x);
	} else {
		tmp = tan(z) - -x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-13)
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	elseif (Float64(y + z) <= 1.0)
		tmp = Float64(Float64(fma(Float64(z * z), 0.3333333333333333, 1.0) * z) - Float64(tan(a) - x));
	else
		tmp = Float64(tan(z) - Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-13], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1.0], N[(N[(N[(N[(z * z), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * z), $MachinePrecision] - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2.0000000000000001e-13

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 43.4

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

   7. Applied rewrites 43.4%

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

   if -2.0000000000000001e-13 < (+.f64 y z) < 1

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 98.5%

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

   if 1 < (+.f64 y z)

   1. Initial program 70.3%

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

      1. lift-+.f64 N/A

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

      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 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 42.1

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 30.7

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

   10. Applied rewrites 30.7%

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

   12. Applied rewrites 30.7%

       \[\leadsto \color{blue}{\tan z - \left(-x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return tan((z + y)) - -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 = tan((z + y)) - -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.5

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 48.2

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

7. Applied rewrites 48.2%

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

Alternative 14: 41.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return tan(z) - -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 = tan(z) - -x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.5

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 48.2

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

7. Applied rewrites 48.2%

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

8. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-cos.f64 39.4

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

10. Applied rewrites 39.4%

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

12. Applied rewrites 39.4%

    \[\leadsto \color{blue}{\tan z - \left(-x\right)} \]
13. Add Preprocessing

Alternative 15: 32.6% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(-x\right) \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double a) {
	return -1.0 * -x;
}

Python

def code(x, y, z, a):
	return -1.0 * -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(-1.0 * (-x)), $MachinePrecision]

Derivation

1. Initial program 78.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 78.5

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} - \frac{\sin a}{\cos a}}{x} - 1\right)\right)} \]
6. 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. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 78.3%

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

8. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
9. Step-by-step derivation

10. Applied rewrites 33.3%

    \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))