tan-example (used to crash)

Average Error: 13.5 → 0.2

Time: 30.8s

Precision: binary64

Cost: 39232

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (+ x (- (* (+ (tan y) (tan z)) (/ -1.0 (fma (tan y) (tan z) -1.0))) (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) {
	return x + (((tan(y) + tan(z)) * (-1.0 / fma(tan(y), tan(z), -1.0))) - 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)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) * Float64(-1.0 / fma(tan(y), tan(z), -1.0))) - 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_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.5

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

2. Applied egg-rr 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   associate-*r/ [=>] 0.2	\[ x + \left(\color{blue}{\frac{\left(\tan y + \tan z\right) \cdot 1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   *-rgt-identity [=>] 0.2	\[ x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]

4. Applied egg-rr 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{-1 + \tan y \cdot \tan z}, -\tan a\right) \]
   fma-udef [=>] 0.2	\[ x + \color{blue}{\left(\left(\tan y + \tan z\right) \cdot \frac{-1}{-1 + \tan y \cdot \tan z} + \left(-\tan a\right)\right)} \]
   unsub-neg [=>] 0.2	\[ x + \color{blue}{\left(\left(\tan y + \tan z\right) \cdot \frac{-1}{-1 + \tan y \cdot \tan z} - \tan a\right)} \]
   +-commutative [=>] 0.2	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{-1}{\color{blue}{\tan y \cdot \tan z + -1}} - \tan a\right) \]
   fma-def [=>] 0.2	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{-1}{\color{blue}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}} - \tan a\right) \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	32832

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

Alternative 2
Error	7.1
Cost	26569

\[\begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{-11} \lor \neg \left(a \leq 2.4 \cdot 10^{-11}\right):\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]

Alternative 3
Error	19.4
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -4 \cdot 10^{-11} \lor \neg \left(\tan a \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 4
Error	25.7
Cost	20041

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-10} \lor \neg \left(t_0 \leq 0.0002\right):\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;z - \left(\tan a - x\right)\\ \end{array} \]

Alternative 5
Error	19.5
Cost	13252

\[\begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 6
Error	13.5
Cost	13248

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

Alternative 7
Error	31.6
Cost	6720

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

Alternative 8
Error	43.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023237 
(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))))