?

Average Error: 13.2 → 0.2
Time: 31.0s
Precision: binary64
Cost: 39104

?

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x - \left(\tan a + \frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, \tan z, -1\right)}\right) \]
(FPCore (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)))))
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)));
}

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

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]

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)

Error?

Derivation?

  1. Initial program 13.2

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

  2. Applied egg-rr0.2

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

  3. Simplified0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
    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-rr0.2

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

  5. Applied egg-rr0.2

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

  6. Simplified0.2

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

    [Start]0.2

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

    fma-udef [=>]0.2

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

    unsub-neg [=>]0.2

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

    associate-*r/ [=>]0.2

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

    *-commutative [<=]0.2

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

    neg-mul-1 [<=]0.2

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

    distribute-neg-in [=>]0.2

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

    unsub-neg [=>]0.2

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

  7. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost32832
\[x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]
Alternative 2
Error6.8
Cost26697
\[\begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -1.42 \lor \neg \left(a \leq 7 \cdot 10^{-31}\right):\\ \;\;\;\;x + \mathsf{fma}\left(t_0, 1, -\tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t_0}{1 - \tan y \cdot \tan z} - a\right)\\ \end{array} \]
Alternative 3
Error12.9
Cost26112
\[x + \mathsf{fma}\left(\tan y + \tan z, 1, -\tan a\right) \]
Alternative 4
Error13.2
Cost13504
\[x + \frac{1}{\frac{1}{\tan \left(y + z\right) - \tan a}} \]
Alternative 5
Error19.3
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -0.00084 \lor \neg \left(a \leq 0.026\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
Alternative 6
Error13.2
Cost13248
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
Alternative 7
Error29.0
Cost7113
\[\begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+27} \lor \neg \left(a \leq 29\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
Alternative 8
Error35.1
Cost6985
\[\begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+27} \lor \neg \left(a \leq 29\right):\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - a\right)\\ \end{array} \]
Alternative 9
Error40.2
Cost6852
\[\begin{array}{l} \mathbf{if}\;y \leq -0.0024:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]
Alternative 10
Error43.8
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023057 
(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))))