tan-example (used to crash)

Average Error: 13.3 → 0.2

Time: 33.0s

Precision: binary64

Cost: 32960

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

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (/ (tan y) (/ 1.0 (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 + (((tan(y) + tan(z)) / (1.0 - (tan(y) / (1.0 / tan(z))))) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

↓

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) / (1.0d0 / tan(z))))) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

↓

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) / (1.0 / Math.tan(z))))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

↓

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) / (1.0 / math.tan(z))))) - math.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(1.0 - Float64(tan(y) / Float64(1.0 / tan(z))))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

↓

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) / (1.0 / tan(z))))) - 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[(1.0 - N[(N[Tan[y], $MachinePrecision] / N[(1.0 / N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.3

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

2. Applied egg-rr 0.2

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

3. Simplified 0.2

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   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-rr 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan y \cdot \sin z}{\cos z}} - \tan a\right) \]
   *-commutative [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \frac{\color{blue}{\sin z \cdot \tan y}}{\cos z}} - \tan a\right) \]
   associate-/l* [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\sin z}{\frac{\cos z}{\tan y}}}} - \tan a\right) \]

6. Applied egg-rr 0.2

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

7. Simplified 0.2

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

   [Start] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \sin z \cdot \frac{\tan y}{\cos z}} - \tan a\right) \]
   *-commutative [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan y}{\cos z} \cdot \sin z}} - \tan a\right) \]
   associate-*l/ [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan y \cdot \sin z}{\cos z}}} - \tan a\right) \]
   associate-/l* [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan y}{\frac{\cos z}{\sin z}}}} - \tan a\right) \]

8. Applied egg-rr 20.1

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan y}{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\tan z}\right)} - 1}}} - \tan a\right) \]

9. Simplified 0.2

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

   [Start] 20.1	\[ x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan y}{e^{\mathsf{log1p}\left(\frac{1}{\tan z}\right)} - 1}} - \tan a\right) \]
   expm1-def [=>] 20.1	\[ x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan y}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan z}\right)\right)}}} - \tan a\right) \]
   expm1-log1p [=>] 0.2	\[ x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan y}{\color{blue}{\frac{1}{\tan z}}}} - \tan a\right) \]

10. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.7
Cost	39881

\[\begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-14} \lor \neg \left(\tan a \leq 0.14\right):\\ \;\;\;\;x + \left(\frac{t_0}{1 - \frac{\tan y}{z \cdot -0.3333333333333333 + \frac{1}{z}}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{1 - \tan y \cdot \tan z}{t_0}}\\ \end{array} \]

Alternative 2
Error	7.9
Cost	39496

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 0.14:\\ \;\;\;\;x + \frac{1}{\frac{1 - \tan y \cdot \tan z}{\tan y + \tan z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 3
Error	7.9
Cost	39368

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;\tan a \leq 0.14:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	32832

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

Alternative 5
Error	19.5
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.36 \lor \neg \left(a \leq 0.00065\right):\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 6
Error	13.5
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-11}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 7
Error	13.5
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 8
Error	13.3
Cost	13248

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

Alternative 9
Error	25.8
Cost	6985

\[\begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+14} \lor \neg \left(a \leq 35\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 10
Error	31.8
Cost	6857

\[\begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+14} \lor \neg \left(a \leq 4.1 \cdot 10^{-16}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]

Alternative 11
Error	37.1
Cost	6592

\[x - \tan a \]

Alternative 12
Error	43.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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))))