tan-example (used to crash)

Average Accuracy: 78.9% → 99.7%

Time: 37.5s

Precision: binary64

Cost: 39296

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

Math

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

↓

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

FPCore

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

↓

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

Derivation

1. Initial program 78.9%

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

2. Applied egg-rr 99.7%

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

   [Start] 78.9	\[ x + \left(\tan \left(y + z\right) - \tan a\right) \]
   tan-sum [=>] 99.7	\[ x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   div-inv [=>] 99.7	\[ x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]

3. Simplified 99.7%

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

   [Start] 99.7	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   associate-*r/ [=>] 99.7	\[ x + \left(\color{blue}{\frac{\left(\tan y + \tan z\right) \cdot 1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   *-rgt-identity [=>] 99.7	\[ x + \left(\frac{\color{blue}{\tan y + \tan z}}{1 - \tan y \cdot \tan z} - \tan a\right) \]

4. Applied egg-rr 99.7%

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

   [Start] 99.7	\[ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   div-inv [=>] 99.7	\[ x + \left(\color{blue}{\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   fma-neg [=>] 99.7	\[ x + \color{blue}{\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\tan a\right)} \]
   frac-2neg [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \color{blue}{\frac{-1}{-\left(1 - \tan y \cdot \tan z\right)}}, -\tan a\right) \]
   metadata-eval [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{\color{blue}{-1}}{-\left(1 - \tan y \cdot \tan z\right)}, -\tan a\right) \]
   neg-sub0 [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{\color{blue}{0 - \left(1 - \tan y \cdot \tan z\right)}}, -\tan a\right) \]
   metadata-eval [<=] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{\color{blue}{\log 1} - \left(1 - \tan y \cdot \tan z\right)}, -\tan a\right) \]
   associate--r- [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{\color{blue}{\left(\log 1 - 1\right) + \tan y \cdot \tan z}}, -\tan a\right) \]
   metadata-eval [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{\left(\color{blue}{0} - 1\right) + \tan y \cdot \tan z}, -\tan a\right) \]
   metadata-eval [=>] 99.7	\[ x + \mathsf{fma}\left(\tan y + \tan z, \frac{-1}{\color{blue}{-1} + \tan y \cdot \tan z}, -\tan a\right) \]

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	88.4%
Cost	39368

\[\begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\left(x + 1\right) + \left(\tan \left(y + z\right) + \left(-1 - \tan a\right)\right)\\ \mathbf{elif}\;\tan a \leq 0.0435:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	32832

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

Alternative 3
Accuracy	68.6%
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-14} \lor \neg \left(\tan a \leq 0.05\right):\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 4
Accuracy	68.6%
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-14} \lor \neg \left(\tan a \leq 0.05\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 5
Accuracy	57.7%
Cost	19785

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-14} \lor \neg \left(\tan a \leq 10^{-83}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 6
Accuracy	79.2%
Cost	19648

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

Alternative 7
Accuracy	78.9%
Cost	13248

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

Alternative 8
Accuracy	78.8%
Cost	13248

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

Alternative 9
Accuracy	41.7%
Cost	6592

\[x - \tan a \]

Alternative 10
Accuracy	31.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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))))