tan-example

Average Error: 13.6 → 0.2

Time: 26.6s

Precision: binary64

Cost: 39360

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

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

Math

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

↓

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

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) (/ (cos z) (sin 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) / (cos(z) / sin(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) / (cos(z) / sin(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) / (Math.cos(z) / Math.sin(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) / (math.cos(z) / math.sin(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(cos(z) / sin(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) / (cos(z) / sin(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[(N[Cos[z], $MachinePrecision] / N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.6

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

2. Simplified 13.6

   \[\leadsto \color{blue}{x - \left(\tan a - \tan \left(y + z\right)\right)} \]
   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)))): 24 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, 24 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-rr 0.2

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

4. Simplified 0.2

   \[\leadsto x - \left(\tan a - \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}}\right) \]
   Proof
   (/.f64 (+.f64 (tan.f64 y) (tan.f64 z)) (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (tan.f64 y) (tan.f64 z)) 1)) (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (tan.f64 y) (tan.f64 z)) (/.f64 1 (-.f64 1 (*.f64 (tan.f64 y) (tan.f64 z)))))): 8 points increase in error, 8 points decrease in error

5. Applied egg-rr 0.2

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

6. Simplified 0.2

   \[\leadsto x - \left(\tan a - \frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan y}{\frac{\cos z}{\sin z}}}}\right) \]
   Proof
   (/.f64 (tan.f64 y) (/.f64 (cos.f64 z) (sin.f64 z))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (tan.f64 y) (sin.f64 z)) (cos.f64 z))): 37 points increase in error, 35 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 z) (tan.f64 y))) (cos.f64 z)): 0 points increase in error, 0 points decrease in error

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.6
Cost	39880

\[\begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(\frac{t_0}{1 - \frac{\tan y}{z \cdot -0.3333333333333333 + \frac{1}{z}}} - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\tan a \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{t_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	8.3
Cost	39368

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;\sqrt{{\left(x + t_0\right)}^{2}} - \tan a\\ \mathbf{elif}\;\tan a \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	32960

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

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	25.6
Cost	26184

\[\begin{array}{l} t_0 := \sqrt{x \cdot x} - \tan a\\ \mathbf{if}\;\tan a \leq -0.002:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	13.6
Cost	13248

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

Alternative 7
Error	31.5
Cost	6720

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

Alternative 8
Error	43.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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))))