tan-example (used to crash)

Average Error: 13.0 → 0.2

Time: 28.6s

Precision: binary64

Cost: 52608

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

Math

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

↓

\[\begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\left(\frac{-1}{-1 + {t_0}^{2}} \cdot \left(\tan y + \tan z\right)\right) \cdot \left(t_0 + 1\right) - \tan a\right) \end{array} \]

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))))
   (+
    x
    (-
     (* (* (/ -1.0 (+ -1.0 (pow t_0 2.0))) (+ (tan y) (tan z))) (+ t_0 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) {
	double t_0 = tan(y) * tan(z);
	return x + ((((-1.0 / (-1.0 + pow(t_0, 2.0))) * (tan(y) + tan(z))) * (t_0 + 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
    real(8) :: t_0
    t_0 = tan(y) * tan(z)
    code = x + (((((-1.0d0) / ((-1.0d0) + (t_0 ** 2.0d0))) * (tan(y) + tan(z))) * (t_0 + 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) {
	double t_0 = Math.tan(y) * Math.tan(z);
	return x + ((((-1.0 / (-1.0 + Math.pow(t_0, 2.0))) * (Math.tan(y) + Math.tan(z))) * (t_0 + 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):
	t_0 = math.tan(y) * math.tan(z)
	return x + ((((-1.0 / (-1.0 + math.pow(t_0, 2.0))) * (math.tan(y) + math.tan(z))) * (t_0 + 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)
	t_0 = Float64(tan(y) * tan(z))
	return Float64(x + Float64(Float64(Float64(Float64(-1.0 / Float64(-1.0 + (t_0 ^ 2.0))) * Float64(tan(y) + tan(z))) * Float64(t_0 + 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)
	t_0 = tan(y) * tan(z);
	tmp = x + ((((-1.0 / (-1.0 + (t_0 ^ 2.0))) * (tan(y) + tan(z))) * (t_0 + 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_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(-1.0 / N[(-1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 13.0

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

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
   (+.f64 x (-.f64 (/.f64 (+.f64 (tan.f64 y) (tan.f64 z)) (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))) (tan.f64 a))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (-.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (tan.f64 y) (tan.f64 z)) 1)) (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))) (tan.f64 a))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (tan.f64 y) (tan.f64 z)) (/.f64 1 (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))))) (tan.f64 a))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	52480

\[\begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\left(t_0 + 1\right) \cdot \frac{\tan y + \tan z}{1 - {t_0}^{2}} - \tan a\right) \end{array} \]

Alternative 2
Error	0.2
Cost	32832

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

Alternative 3
Error	25.6
Cost	19913

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -0.01 \lor \neg \left(\tan a \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 4
Error	25.6
Cost	19912

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x - \frac{\sin a}{\cos a}\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]

Alternative 5
Error	26.3
Cost	19785

\[\begin{array}{l} \mathbf{if}\;\tan a \leq -0.05 \lor \neg \left(\tan a \leq 0.32\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 6
Error	13.0
Cost	13248

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

Alternative 7
Error	37.2
Cost	6592

\[x - \tan a \]

Alternative 8
Error	44.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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))))