Average Error: 13.4 → 0.2
Time: 45.8s
Precision: binary64
Cost: 39168
\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
↓
\[x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)} - \tan a\right) \]
(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))))
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));
}

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)) / fma(tan(y), Float64(-tan(z)), 1.0)) - tan(a)))
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[(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]

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)

Error

Derivation

  1. Initial program 13.4

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

  2. Simplified13.4

    \[\leadsto \color{blue}{x - \left(\tan a - \tan \left(y + z\right)\right)} \]
    Proof
    (-.f64 x (-.f64 (tan.f64 a) (tan.f64 (+.f64 y z)))): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 x (tan.f64 a)) (tan.f64 (+.f64 y z)))): 22 points increase in error, 6 points decrease in error
    (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 x (neg.f64 (tan.f64 a)))) (tan.f64 (+.f64 y z))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 x (+.f64 (neg.f64 (tan.f64 a)) (tan.f64 (+.f64 y z))))): 6 points increase in error, 22 points decrease in error
    (+.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (tan.f64 (+.f64 y z)) (neg.f64 (tan.f64 a))))): 0 points increase in error, 0 points decrease in error
    (+.f64 x (Rewrite<= sub-neg_binary64 (-.f64 (tan.f64 (+.f64 y z)) (tan.f64 a)))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr13.4

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

  4. Applied egg-rr0.3

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

  5. Applied egg-rr0.2

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

  6. Final simplification0.2

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

Alternatives

Alternative 1
Error7.6
Cost33096
\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -7.658084313671034 \cdot 10^{+20}:\\ \;\;\;\;x + \left(\log \left(e^{t_0}\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 3.065002401447037 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \log \left(e^{\tan a - t_0}\right)\\ \end{array} \]
Alternative 2
Error0.2
Cost32960
\[x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - \tan a\right) \]
Alternative 3
Error0.3
Cost32832
\[\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - \tan a\right) \]
Alternative 4
Error0.2
Cost32832
\[x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]
Alternative 5
Error7.6
Cost26824
\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -7.658084313671034 \cdot 10^{+20}:\\ \;\;\;\;x + \left(\log \left(e^{t_0}\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 3.065002401447037 \cdot 10^{-8}:\\ \;\;\;\;\left(x - a\right) + \frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}}\\ \mathbf{else}:\\ \;\;\;\;x - \log \left(e^{\tan a - t_0}\right)\\ \end{array} \]
Alternative 6
Error7.6
Cost26696
\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -7.658084313671034 \cdot 10^{+20}:\\ \;\;\;\;x + \left(\log \left(e^{t_0}\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 3.065002401447037 \cdot 10^{-8}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \log \left(e^{\tan a - t_0}\right)\\ \end{array} \]
Alternative 7
Error13.4
Cost20032
\[x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right) \]
Alternative 8
Error13.4
Cost13248
\[\left(x + \tan \left(y + z\right)\right) - \tan a \]
Alternative 9
Error13.4
Cost13248
\[x + \left(\tan \left(y + z\right) - \tan a\right) \]
Alternative 10
Error32.1
Cost6720
\[x + \tan \left(y + z\right) \]
Alternative 11
Error43.7
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z a)
  :name "tan-example"
  :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))))