tan-example (used to crash)

Average Error: 13.4 → 13.4

Time: 50.4s

Precision: binary64

Cost: 19840

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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 + y)) + (sin(a) / -cos(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));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 13.4

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

2. Applied egg-rr 13.4

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

3. Taylor expanded in a around inf 13.4

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

4. Applied egg-rr 13.4

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

5. Simplified 13.4

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

   [Start] 13.4	\[ x + \left(\left(\tan \left(y + z\right) + \left(1 - \frac{\sin a}{\cos a}\right)\right) - 1\right) \]
   rational.json-simplify-1 [=>] 13.4	\[ x + \left(\color{blue}{\left(\left(1 - \frac{\sin a}{\cos a}\right) + \tan \left(y + z\right)\right)} - 1\right) \]
   rational.json-simplify-48 [=>] 13.4	\[ x + \color{blue}{\left(\tan \left(y + z\right) + \left(\left(1 - \frac{\sin a}{\cos a}\right) - 1\right)\right)} \]
   rational.json-simplify-42 [=>] 13.4	\[ x + \left(\tan \left(y + z\right) + \color{blue}{\left(\left(1 - 1\right) - \frac{\sin a}{\cos a}\right)}\right) \]
   metadata-eval [=>] 13.4	\[ x + \left(\tan \left(y + z\right) + \left(\color{blue}{0} - \frac{\sin a}{\cos a}\right)\right) \]
   rational.json-simplify-12 [<=] 13.4	\[ x + \left(\tan \left(y + z\right) + \color{blue}{\left(-\frac{\sin a}{\cos a}\right)}\right) \]
   rational.json-simplify-11 [<=] 13.4	\[ x + \left(\tan \left(y + z\right) + \color{blue}{\frac{\frac{\sin a}{\cos a}}{-1}}\right) \]
   rational.json-simplify-46 [<=] 13.4	\[ x + \left(\tan \left(y + z\right) + \color{blue}{\frac{\sin a}{\cos a \cdot -1}}\right) \]
   rational.json-simplify-8 [<=] 13.4	\[ x + \left(\tan \left(y + z\right) + \frac{\sin a}{\color{blue}{-\cos a}}\right) \]
   rational.json-simplify-1 [=>] 13.4	\[ x + \left(\tan \color{blue}{\left(z + y\right)} + \frac{\sin a}{-\cos a}\right) \]

6. Final simplification 13.4

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

Alternatives

Alternative 1
Error	25.7
Cost	13384

\[\begin{array}{l} t_0 := x - \frac{\sin a}{\cos a}\\ \mathbf{if}\;a \leq -0.011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.75:\\ \;\;\;\;x + \left(\tan \left(z + y\right) + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.4
Cost	13248

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

Alternative 3
Error	31.7
Cost	6720

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

Alternative 4
Error	43.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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))))