tan-example (used to crash)

Average Error: 13.1 → 7.1

Time: 38.2s

Precision: binary64

Cost: 52552

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

Math

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

↓

\[\begin{array}{l} t_0 := x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\cos z \cdot \sin y + \cos y \cdot \sin z}{\cos z \cdot \cos y - \sin z \cdot \sin y} + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan (+ y z)) (tan a)))))
   (if (<= a -2.1e-6)
     t_0
     (if (<= a 3.4e-16)
       (+
        (/
         (+ (* (cos z) (sin y)) (* (cos y) (sin z)))
         (- (* (cos z) (cos y)) (* (sin z) (sin y))))
        x)
       t_0))))

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 = x + (tan((y + z)) - tan(a));
	double tmp;
	if (a <= -2.1e-6) {
		tmp = t_0;
	} else if (a <= 3.4e-16) {
		tmp = (((cos(z) * sin(y)) + (cos(y) * sin(z))) / ((cos(z) * cos(y)) - (sin(z) * sin(y)))) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = x + (tan((y + z)) - tan(a))
    if (a <= (-2.1d-6)) then
        tmp = t_0
    else if (a <= 3.4d-16) then
        tmp = (((cos(z) * sin(y)) + (cos(y) * sin(z))) / ((cos(z) * cos(y)) - (sin(z) * sin(y)))) + x
    else
        tmp = t_0
    end if
    code = tmp
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 = x + (Math.tan((y + z)) - Math.tan(a));
	double tmp;
	if (a <= -2.1e-6) {
		tmp = t_0;
	} else if (a <= 3.4e-16) {
		tmp = (((Math.cos(z) * Math.sin(y)) + (Math.cos(y) * Math.sin(z))) / ((Math.cos(z) * Math.cos(y)) - (Math.sin(z) * Math.sin(y)))) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, a):
	t_0 = x + (math.tan((y + z)) - math.tan(a))
	tmp = 0
	if a <= -2.1e-6:
		tmp = t_0
	elif a <= 3.4e-16:
		tmp = (((math.cos(z) * math.sin(y)) + (math.cos(y) * math.sin(z))) / ((math.cos(z) * math.cos(y)) - (math.sin(z) * math.sin(y)))) + x
	else:
		tmp = t_0
	return tmp

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(x + Float64(tan(Float64(y + z)) - tan(a)))
	tmp = 0.0
	if (a <= -2.1e-6)
		tmp = t_0;
	elseif (a <= 3.4e-16)
		tmp = Float64(Float64(Float64(Float64(cos(z) * sin(y)) + Float64(cos(y) * sin(z))) / Float64(Float64(cos(z) * cos(y)) - Float64(sin(z) * sin(y)))) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

↓

function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan((y + z)) - tan(a));
	tmp = 0.0;
	if (a <= -2.1e-6)
		tmp = t_0;
	elseif (a <= 3.4e-16)
		tmp = (((cos(z) * sin(y)) + (cos(y) * sin(z))) / ((cos(z) * cos(y)) - (sin(z) * sin(y)))) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
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[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.1e-6], t$95$0, If[LessEqual[a, 3.4e-16], N[(N[(N[(N[(N[Cos[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[z], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0999999999999998e-6 or 3.4e-16 < a

   1. Initial program 13.4

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

   if -2.0999999999999998e-6 < a < 3.4e-16

   1. Initial program 12.7

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

   2. Taylor expanded in a around 0 12.9

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

   3. Simplified 12.9

      \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + x} \]
      Proof

      [Start] 12.9	\[ \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)} + x \]
      rational_best.json-simplify-1 [=>] 12.9	\[ \frac{\sin \color{blue}{\left(z + y\right)}}{\cos \left(y + z\right)} + x \]
      rational_best.json-simplify-1 [=>] 12.9	\[ \frac{\sin \left(z + y\right)}{\cos \color{blue}{\left(z + y\right)}} + x \]

   4. Applied egg-rr 12.6

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

   5. Applied egg-rr 0.4

      \[\leadsto \frac{\cos z \cdot \sin y + \cos y \cdot \sin z}{\color{blue}{\cos z \cdot \cos y - \sin z \cdot \sin y}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\cos z \cdot \sin y + \cos y \cdot \sin z}{\cos z \cdot \cos y - \sin z \cdot \sin y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	31.5
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\tan \left(y + z\right) + \left(x + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{\cos y} + x\\ \end{array} \]

Alternative 2
Error	31.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\tan \left(y + z\right) + \left(x + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin z}{\cos z} + x\\ \end{array} \]

Alternative 3
Error	25.4
Cost	13384

\[\begin{array}{l} t_0 := x - \frac{\sin a}{\cos a}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.15:\\ \;\;\;\;\tan \left(y + z\right) + \left(x + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.1
Cost	13248

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

Alternative 5
Error	31.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.2:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.56:\\ \;\;\;\;\tan \left(y + z\right) + \left(x + \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	43.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023089 
(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))))