tan-example (used to crash)

Average Error: 12.8 → 0.2

Time: 30.6s

Precision: binary64

Cost: 39104

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

Math

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

↓

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

FPCore

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

↓

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

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(a) + ((tan(y) + tan(z)) / fma(tan(y), tan(z), -1.0)));
}

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(a) + Float64(Float64(tan(y) + tan(z)) / fma(tan(y), tan(z), -1.0))))
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[a], $MachinePrecision] + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.8

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

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

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

6. Final simplification 0.2

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

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	19.4
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 0.0042\right):\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(x + \tan \left(y + z\right)\right) + \left(-1 - a\right)\right)\\ \end{array} \]

Alternative 3
Error	12.6
Cost	19776

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

Alternative 4
Error	12.8
Cost	13248

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

Alternative 5
Error	25.2
Cost	7369

\[\begin{array}{l} \mathbf{if}\;a \leq -0.31 \lor \neg \left(a \leq 0.057\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(x + \tan \left(y + z\right)\right) + \left(-1 - a\right)\right)\\ \end{array} \]

Alternative 6
Error	25.7
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y + z \leq -50 \lor \neg \left(y + z \leq 0.001\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]

Alternative 7
Error	25.2
Cost	7113

\[\begin{array}{l} \mathbf{if}\;a \leq -0.05 \lor \neg \left(a \leq 0.045\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 8
Error	37.2
Cost	6592

\[x - \tan a \]

Alternative 9
Error	43.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023059 
(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))))