tan-example (used to crash)

Average Error: 13.0 → 0.2

Time: 32.7s

Precision: binary64

Cost: 65024

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

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (fma (tan z) (tan y) -1.0)))
   (+ x (fma (tan z) (/ -1.0 t_0) (- (/ (- (tan y)) t_0) (tan 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) {
	double t_0 = fma(tan(z), tan(y), -1.0);
	return x + fma(tan(z), (-1.0 / t_0), ((-tan(y) / t_0) - tan(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)
	t_0 = fma(tan(z), tan(y), -1.0)
	return Float64(x + fma(tan(z), Float64(-1.0 / t_0), Float64(Float64(Float64(-tan(y)) / t_0) - 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]

↓

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[Tan[z], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision] + N[(N[((-N[Tan[y], $MachinePrecision]) / t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 13.0

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

2. Applied egg-rr 0.2

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Simplified 0.2

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

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

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.0
Cost	39368

\[\begin{array}{l} t_0 := \tan \left(z + y\right)\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\left(-1 - \tan a\right) + \left(1 + t_0\right)\right)\\ \mathbf{elif}\;\tan a \leq 10^{-25}:\\ \;\;\;\;x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	39232

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

Alternative 3
Error	0.2
Cost	32960

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

Alternative 4
Error	0.2
Cost	32832

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

Alternative 5
Error	19.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z + y \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \end{array} \]

Alternative 6
Error	19.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z + y \leq 5 \cdot 10^{-32}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \end{array} \]

Alternative 7
Error	13.0
Cost	13248

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

Alternative 8
Error	31.4
Cost	6720

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

Alternative 9
Error	43.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023066 
(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))))