tan-example (used to crash)

Average Error: 13.4 → 0.2

Time: 34.2s

Precision: binary64

Cost: 39360

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

Math

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

↓

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

FPCore

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

↓

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

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 (((tan(y) + tan(z)) / (1.0 - ((tan(z) * sin(y)) / cos(y)))) - tan(a)) + x;
}

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 = (((tan(y) + tan(z)) / (1.0d0 - ((tan(z) * sin(y)) / cos(y)))) - tan(a)) + x
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 (((Math.tan(y) + Math.tan(z)) / (1.0 - ((Math.tan(z) * Math.sin(y)) / Math.cos(y)))) - Math.tan(a)) + x;
}

Python

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

↓

def code(x, y, z, a):
	return (((math.tan(y) + math.tan(z)) / (1.0 - ((math.tan(z) * math.sin(y)) / math.cos(y)))) - math.tan(a)) + x

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

Derivation

1. Initial program 13.4

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

2. Simplified 13.4

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

   [Start] 13.4	\[ x + \left(\tan \left(y + z\right) - \tan a\right) \]
   associate-+r- [=>] 13.4	\[ \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	39360

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

Alternative 2
Error	0.2
Cost	39296

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

Alternative 3
Error	0.2
Cost	32832

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

Alternative 4
Error	7.0
Cost	26696

\[\begin{array}{l} t_0 := x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{if}\;a \leq -0.0005:\\ \;\;\;\;\log \left(e^{t_0}\right)\\ \mathbf{elif}\;a \leq 0.02:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.8
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -0.005 \lor \neg \left(\tan a \leq 0.02\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 6
Error	25.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 7
Error	13.4
Cost	13248

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

Alternative 8
Error	28.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0105:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{elif}\;a \leq 120:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]

Alternative 9
Error	28.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;a \leq -0.0105:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{elif}\;a \leq 120:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - \tan a\right)\\ \end{array} \]

Alternative 10
Error	40.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]

Alternative 11
Error	43.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(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))))