tan-example

Average Error: 13.4 → 0.2

Time: 33.4s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

FPCore

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

↓

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

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

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

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(a) - ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))))

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(tan(a) - Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z))))))
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(a) - ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))));
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[Tan[a], $MachinePrecision] - N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.4

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

2. Simplified 13.4

   \[\leadsto \color{blue}{x - \left(\tan a - \tan \left(y + z\right)\right)} \]
   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)))): 23 points increase in error, 10 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))))): 10 points increase in error, 23 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

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.2
Cost	45832

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-10}:\\ \;\;\;\;\left(x + \frac{\tan y + \tan z}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)}\right) - a\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan a - t_0\right)\right)\\ \end{array} \]

Alternative 2
Error	7.2
Cost	39496

\[\begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-10}:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{log1p}\left(\mathsf{expm1}\left(\tan a - t_0\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	32832

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

Alternative 4
Error	25.7
Cost	26184

\[\begin{array}{l} t_0 := x - \frac{\sin a}{\cos a}\\ \mathbf{if}\;\tan a \leq -0.03:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\tan a \leq 10^{-10}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.9
Cost	26184

\[\begin{array}{l} t_0 := \tan y + \left(x - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-35}:\\ \;\;\;\;x - \left(a - \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.9
Cost	26184

\[\begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-35}:\\ \;\;\;\;x - \left(a - \tan \left(y + z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	13.4
Cost	13248

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

Alternative 8
Error	13.4
Cost	13248

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

Alternative 9
Error	13.4
Cost	13248

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

Alternative 10
Error	26.3
Cost	7240

\[\begin{array}{l} t_0 := x + \tan \left(y + z\right)\\ \mathbf{if}\;y + z \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y + z \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	40.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -193921.32439130233:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \tan a\right)\\ \end{array} \]

Alternative 12
Error	43.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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))))