tan-example (used to crash)

Average Error: 13.5 → 0.2

Time: 44.8s

Precision: binary64

Cost: 39424

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (- (+ (tan z) (tan y))) (- (/ (* (tan z) (sin y)) (cos y)) 1.0))
   (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(z) + tan(y)) / (((tan(z) * sin(y)) / cos(y)) - 1.0)) - 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(z) + tan(y)) / (((tan(z) * sin(y)) / cos(y)) - 1.0d0)) - 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(z) + Math.tan(y)) / (((Math.tan(z) * Math.sin(y)) / Math.cos(y)) - 1.0)) - 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(z) + math.tan(y)) / (((math.tan(z) * math.sin(y)) / math.cos(y)) - 1.0)) - 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(-Float64(tan(z) + tan(y))) / Float64(Float64(Float64(tan(z) * sin(y)) / cos(y)) - 1.0)) - 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(z) + tan(y)) / (((tan(z) * sin(y)) / cos(y)) - 1.0)) - 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[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]) / N[(N[(N[(N[Tan[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.5

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

2. Applied egg-rr 0.2

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

3. Taylor expanded in y around inf 0.2

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

4. Simplified 0.2

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

5. Applied egg-rr 0.2

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

Alternatives

Alternative 1
Error	7.8
Cost	39432

\[\begin{array}{l} t_0 := \left(x - \frac{\tan y + \tan z}{-1}\right) - \tan a\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\tan a \leq 0.102:\\ \;\;\;\;\frac{-\left(\tan z + \tan y\right)}{\tan z \cdot \tan y - 1} + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	32896

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

Alternative 3
Error	0.2
Cost	32832

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

Alternative 4
Error	7.1
Cost	26824

\[\begin{array}{l} \mathbf{if}\;a \leq -2:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\frac{-\left(\tan z + \tan y\right)}{\tan z \cdot \tan y - 1} + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\tan y + \tan z}{-1}\right) - \tan a\\ \end{array} \]

Alternative 5
Error	13.2
Cost	19776

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

Alternative 6
Error	19.8
Cost	13384

\[\begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;a \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 270000:\\ \;\;\;\;\left(\tan \left(y + z\right) + \left(-a\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	13.6
Cost	13384

\[\begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	13.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]

Alternative 9
Error	13.5
Cost	13248

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

Alternative 10
Error	13.5
Cost	13248

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

Alternative 11
Error	32.3
Cost	7252

\[\begin{array}{l} t_0 := x - \tan a\\ t_1 := \tan y + x\\ t_2 := \tan z + x\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	25.7
Cost	7176

\[\begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 270000:\\ \;\;\;\;\left(\tan \left(y + z\right) + \left(-a\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	25.9
Cost	6984

\[\begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 760000:\\ \;\;\;\;\tan \left(y + z\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	31.9
Cost	6856

\[\begin{array}{l} t_0 := \tan y + x\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-5}:\\ \;\;\;\;\tan z + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	37.8
Cost	6592

\[\tan y + x \]

Alternative 16
Error	43.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(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))))