tan-example (used to crash)

Average Accuracy: 80.2% → 99.7%

Time: 37.3s

Precision: binary64

Cost: 45888

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (-
  x
  (-
   (tan a)
   (*
    (fma (sin y) (/ 1.0 (cos y)) (tan z))
    (/ 1.0 (- 1.0 (* (tan z) (tan y))))))))

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) - (fma(sin(y), (1.0 / cos(y)), tan(z)) * (1.0 / (1.0 - (tan(z) * tan(y))))));
}

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(fma(sin(y), Float64(1.0 / cos(y)), tan(z)) * Float64(1.0 / Float64(1.0 - Float64(tan(z) * tan(y)))))))
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[Sin[y], $MachinePrecision] * N[(1.0 / N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

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] 80.2	\[ x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Applied egg-rr 99.7%

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

   [Start] 99.7	\[ x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   tan-quot [=>] 99.7	\[ x + \left(\left(\color{blue}{\frac{\sin y}{\cos y}} + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   div-inv [=>] 99.7	\[ x + \left(\left(\color{blue}{\sin y \cdot \frac{1}{\cos y}} + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
   fma-def [=>] 99.7	\[ x + \left(\color{blue}{\mathsf{fma}\left(\sin y, \frac{1}{\cos y}, \tan z\right)} \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	32960

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

Alternative 2
Accuracy	99.7%
Cost	32832

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

Alternative 3
Accuracy	89.3%
Cost	26696

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{1}{1 - \tan z \cdot \tan y} \cdot \left(\tan z + \tan y\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 4
Accuracy	89.3%
Cost	26568

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -4.5 \cdot 10^{-15}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 5
Accuracy	70.6%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.016 \lor \neg \left(a \leq 0.00068\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 6
Accuracy	70.7%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0128:\\ \;\;\;\;\left(x + \tan y\right) - \tan a\\ \mathbf{elif}\;a \leq 0.0009:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]

Alternative 7
Accuracy	80.2%
Cost	13248

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

Alternative 8
Accuracy	50.7%
Cost	6720

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

Alternative 9
Accuracy	31.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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))))