tan-example (used to crash)

Percentage Accurate: 80.8% → 99.7%

Time: 1.2min

Precision: binary64

Cost: 39168

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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

Derivation

1. Initial program 84.0%

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

2. Applied egg-rr 99.8%

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

   [Start] 84.0%	\[ x + \left(\tan \left(y + z\right) - \tan a\right) \]
   tan-sum [=>] 99.8%	\[ x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   div-inv [=>] 99.8%	\[ x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]

3. Simplified 99.8%

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

   [Start] 99.8%	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   associate-*r/ [=>] 99.8%	\[ x + \left(\color{blue}{\frac{\left(\tan y + \tan z\right) \cdot 1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   *-rgt-identity [=>] 99.8%	\[ x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]

4. Applied egg-rr 99.8%

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

   [Start] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   expm1-log1p-u [=>] 94.0%	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]
   expm1-udef [=>] 94.0%	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\tan y \cdot \tan z\right)} - 1\right)}} - \tan a\right) \]
   log1p-udef [=>] 94.0%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(e^{\color{blue}{\log \left(1 + \tan y \cdot \tan z\right)}} - 1\right)} - \tan a\right) \]
   add-exp-log [<=] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(\color{blue}{\left(1 + \tan y \cdot \tan z\right)} - 1\right)} - \tan a\right) \]

5. Simplified 99.8%

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

   [Start] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(\left(1 + \tan y \cdot \tan z\right) - 1\right)} - \tan a\right) \]
   associate--l+ [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(1 + \left(\tan y \cdot \tan z - 1\right)\right)}} - \tan a\right) \]
   fma-neg [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(1 + \color{blue}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}\right)} - \tan a\right) \]
   metadata-eval [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(1 + \mathsf{fma}\left(\tan y, \tan z, \color{blue}{-1}\right)\right)} - \tan a\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto x + \color{blue}{\left(\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(-\tan a\right)\right)} \]
   Step-by-step derivation

   [Start] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{1 - \left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} - \tan a\right) \]
   sub-neg [=>] 99.8%	\[ x + \color{blue}{\left(\frac{\tan y + \tan z}{1 - \left(1 + \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} + \left(-\tan a\right)\right)} \]
   associate--r+ [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{\color{blue}{\left(1 - 1\right) - \mathsf{fma}\left(\tan y, \tan z, -1\right)}} + \left(-\tan a\right)\right) \]
   metadata-eval [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{\color{blue}{0} - \mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(-\tan a\right)\right) \]

7. Simplified 99.8%

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

   [Start] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(-\tan a\right)\right) \]
   sub-neg [<=] 99.8%	\[ x + \color{blue}{\left(\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)} - \tan a\right)} \]
   *-lft-identity [<=] 99.8%	\[ x + \left(\color{blue}{1 \cdot \frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)}} - \tan a\right) \]
   *-lft-identity [=>] 99.8%	\[ x + \left(\color{blue}{\frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)}} - \tan a\right) \]
   sub0-neg [=>] 99.8%	\[ x + \left(\frac{\tan y + \tan z}{\color{blue}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)}} - \tan a\right) \]

8. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	39168

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

Alternative 2
Accuracy	88.3%
Cost	45768

\[\begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan y \cdot \tan z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + \tan \left(y + z\right)\right)}^{2}} - \tan a\\ \end{array} \]

Alternative 3
Accuracy	88.3%
Cost	39368

\[\begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.1:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + \tan \left(y + z\right)\right)}^{2}} - \tan a\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	32832

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

Alternative 5
Accuracy	81.2%
Cost	19648

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

Alternative 6
Accuracy	66.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 7
Accuracy	80.8%
Cost	13248

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

Alternative 8
Accuracy	52.0%
Cost	6720

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

Alternative 9
Accuracy	32.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(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))))