tan-example (used to crash)

Average Error: 13.6 → 0.2

Time: 31.6s

Precision: binary64

Cost: 39360

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

Math

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

↓

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

FPCore

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

↓

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

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 (((tan(y) + tan(z)) / (1.0 - (sin(z) / (cos(z) / tan(y))))) - tan(a)) + x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (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 = (((tan(y) + tan(z)) / (1.0d0 - (sin(z) / (cos(z) / tan(y))))) - tan(a)) + x
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 (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.sin(z) / (Math.cos(z) / Math.tan(y))))) - Math.tan(a)) + x;
}

Python

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

↓

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

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(Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(sin(z) / Float64(cos(z) / tan(y))))) - tan(a)) + x)
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 = (((tan(y) + tan(z)) / (1.0 - (sin(z) / (cos(z) / tan(y))))) - tan(a)) + x;
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[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[z], $MachinePrecision] / N[(N[Cos[z], $MachinePrecision] / N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 13.6

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

2. Simplified 13.6

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

   [Start] 13.6	\[ x + \left(\tan \left(y + z\right) - \tan a\right) \]
   associate-+r- [=>] 13.6	\[ \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

3. Applied egg-rr 0.3

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

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Simplified 0.2

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

   [Start] 0.2	\[ \left(\frac{\tan y + \tan z}{1 - \frac{\tan y \cdot \sin z}{\cos z}} - \tan a\right) + x \]
   *-commutative [<=] 0.2	\[ \left(\frac{\tan y + \tan z}{1 - \frac{\color{blue}{\sin z \cdot \tan y}}{\cos z}} - \tan a\right) + x \]
   associate-/l* [=>] 0.2	\[ \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\sin z}{\frac{\cos z}{\tan y}}}} - \tan a\right) + x \]

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.7
Cost	39496

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-47}:\\ \;\;\;\;x + \left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 2
Error	7.7
Cost	39368

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 10^{-47}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	32832

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

Alternative 4
Error	13.6
Cost	13248

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

Alternative 5
Error	32.2
Cost	6720

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

Alternative 6
Error	43.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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))))